Keywords

3.1 Introduction

Problem-solving has always been a central topic in mathematics education. In the literature, researchers’ exploration of the problem-solving process is usually focused on the entire process of completing the task and solving the problem. Pólya (1994) first introduced the problem-solving process academically in his book, which attracted the attention of mathematics education researchers. Pólya’s mathematical problem-solving model comprises four stages: understanding the problem, proposing a solution, implementing the solution, and checking whether the problem is solved (Pólya, 2011; Pólya, 1944). The process emphasises that learners must clarify the known conditions and the problem objectives, draw graphs, verify and perform other practical processes, and exercise their problem-solving thinking ability by repeating the process. At the end of the twentieth century, Alan Schoenfeld further developed the problem-solving model, proposing that it is not necessarily a smooth linear process, and learners may constantly need to adjust their problem-solving strategies and processes. In his model, problem-solving must incorporate the following processes: analysing and simplifying problems, clarifying problem-solving principles and mechanisms, designing problem-solving strategies, and constructing problem-solving methods from macro to micro perspectives. Difficulties, large or small, may be encountered in the second stage, so learners must go through further exploration and exploration, returning to the problem analysis stage to make adjustments, implement problem-solving strategies, obtain possible results, and verify their feasibility, further generalising feasible results (Schoenfeld, 1985). Therefore, in Schoenfeld’s problem-solving model, the problem-solving process’ analysis and design phases are cyclical processes, emphasising flexibility, and iterative construction.

However, the above models involve solving mathematical problems from the perspective of cognition and subject knowledge skills. Although such models emphasise the transformation and application of mathematical knowledge and skills in completing problem-solving tasks, they do not consider the emotional, psychological, or social processes that learners may experience. For example, it has not been considered whether the learner will be unable to complete the problem-solving process due to their inability to propose a reasonable problem-solving strategy, which affects their confidence in problem-solving. Different from a single mathematical problem-solving situation, multiple students usually participate in the open-ended mathematical problem involved in a collaborative mathematical problem-solving situation. In this process, group members must not only complete tasks to solve problems but also experience social processes such as group work, coordinating problem-solving sub-goals with other members, and listening to others’ opinions. Because of its complex social nature, it becomes difficult to establish a widely recognised collaborative mathematical problem-solving model in the current research field.

Chinese scholars Xu and colleagues proposed using collaborative literacy and its three elements (vision recognition, responsibility sharing, and negotiation and progress) to indicate whether students can work well in groups. Their model emphasises the procedural nature of collaboration and believes that in the process of completing collaborative tasks, learners need to mobilise their meta-cognitive strategies, adjust goals at different stages, make continuous corrections through immediate execution, reflection, and evaluation, and gradually realise the purpose of collaboration (Xu et al., 2020). The model not only emphasises the procedural nature of task completion in the context of collaboration but also macroscopically describes the social process that learners need to participate in the collaboration process. However, completing the task in a collaborative mathematical problem-solving situation requires considering not only the process but also the sociality in the collaborative situation. Therefore, it is necessary to promote more specific and related research rooted in the task completion process to explore the path of collaborative mathematical problem-solving. The negotiation process in group work may be seen as a starting point for related research.

3.2 Literature Review

3.2.1 Students’ Negotiation Process in Groups

Students’ involvement in collaborative group mathematics problem-solving involves not only participation but also negotiation. In a group, everyone may initially have a different understanding of the same problem but then, through discussion, gradually come to understand each other’s views and perspectives. The researchers use the phrase “negotiation of meaning” to conceptualise the social negotiation process through which people negotiate to make themselves understood and understand others. Clarke (2001a, 2001b) described the negotiation process as “a cyclical process of interpretation, reflection, and representation based on shared goals” (p. 35). In the negotiation process, group members gradually understand others’ words and thoughts. When viewed as a negotiation process, the content of group negotiation can illustrate the discussion’s core and the participants’ common goal (e.g., solving a problem).

The existing research usually adopts one of two ways to study the negotiation process. Some researchers have focused on exploring group influences in the negotiation process; that is, information shared, exchanged, and refined within the group in, for example, the discussion process. DeJarnette (2018) saw negotiation as a process in which group members exchange information and behaviour during turn-taking in group activities. On the other hand, some researchers, in addition to focusing on the information shared and exchanged within the group, also pay attention to the degree of individual negotiation in group activities; that is, how each member negotiates or compromises with themself through moment-to-moment discussion. For example, Engle et al. (2014) studied the extent to which an individual’s multiple aspects (knowledge authority, freedom of dialogue, spatial priority) can be compromised in a social context. This study does not view these two perspectives on inquiry negotiation as dichotomous and considers both. By focusing on the negotiation process in collaborative mathematical problem-solving, researchers pay attention to the purpose and topics shared in group discussions and individuals’ participation in and contribution to the group discussion process.

3.2.2 Research on Negotiation Discourse in Mathematics Classrooms

Although peer-led group collaborative learning has a long history in classroom practice, research systematically exploring whether mutual communication among students can promote mathematics learning and the communication mode therein dates from the 1980s (Webb, 1982). Based on 19 published relevant empirical studies, Webb (1989) summarised a simple model between peer communication and mathematics learning, arguing that in group discussions, issues such as high-level interpretation, member feedback quality, and whether to get feedback are all important group interaction factors affecting mathematics learning. Current research on mathematics education promotes the development of mathematical discourse activities by promoting students’ in-depth communication activities in the classroom (Cobb and Bauersfeld, 1995; Inagaki et al., 1998; Moschkovich, 2002; Chan and Sfard, 2020). The dialogues and exchanges in group activities provide rich information for exploring the meaning behind interactive and discourse behaviours from students’ perspectives.

Australian scholars David Clarke and Lihua Xu compared the speech of students in mathematics classrooms in Shanghai, Seoul, and Melbourne and deeply explored the role of language in mathematics classrooms in different cultural backgrounds, finding that whether students have a place to speak in classrooms is a culturally influenced behaviour. For example, in the Seoul classroom, students only spoke in groups and rarely voiced individual comments, whereas there was almost no group-speaking session in Melbourne classrooms. Additionally, the terminology used by teachers and students varied in different classrooms and teaching practices. Although teachers and students had frequent exchanges and conversations in Melbourne classrooms, mathematical terms were used relatively infrequently. In the group speeches in Shanghai and Seoul classrooms, there were two discourse patterns: teacher questioning—collective answering—teacher feedback, and teacher questioning—individual student answering—collective feedback. The mathematical content contained in these two modes of discourse was different, with the former containing less mathematical content and the latter usually including an understanding of basic problem-solving concepts or procedures. Students were given opportunities to strengthen their understanding of concepts and participate in the teaching process during group assessment. Thus, different discourse patterns create different student engagement and learning opportunities (Xu and Clarke, 2019).

Supported by the international Learner’s Perspective Study (LPS) project, mathematics researchers from various countries have researched discourse manifestations in mathematics classrooms based on classroom teaching videos. Chinese scholars have also used videos to analyse the structure of teacher–student discourse and compare the discourse volume in Chinese mathematics classrooms. Early research on teacher–student discourse in Chinese classrooms is usually based on statistical research methods, comparing the number of teachers and students in the classroom and differences in practice. Cao and colleagues analysed the quantity and length of teacher–student discourse in a five-class mathematics classroom for four teachers and concluded that the proportion of teacher and student discourse varied greatly between different mathematics classrooms, although teachers’ discourse in the classroom generally is far greater than students’ (Cao et al., 2008). Dong et al. explored the influence of teacher–student interaction in the classroom and questioned the core of students’ mathematics ability learning and development, taking the discourse interaction between teachers and students as the main research object and the teacher’s questioning as the key starting point. Their study proposed that teachers’ questions not only consciously included teachers’ consideration for achieving curriculum goals but also unconsciously included how to develop students’ thinking skills (Dong et al., 2019). Although the above similar research highlights the importance of teacher–student interaction discourse in the classroom, it also reveals that the teacher discourse’s absolute power in the classroom is dominated by teaching and may ignore peers’ discourses to an extent (Webb, 1989). In fact, students will establish a notable discourse order and culture in a specific classroom environment.

In the context of collaborative mathematics problem-solving, multiple foci occur in student interactions. Firstly, students need to negotiate the subject matter of mathematics, including the tools used in solving problems and the particular language of mathematics, such as facts, procedures, propositions, and so on (Schoenfeld, 1992; Steinbring, 1991). Secondly, students must negotiate norms of social behaviour suited to the mathematics classroom, including teachers’ and students’ rights and duties, to ensure the class’ effective functioning (Brousseau, 1986, 2002). Yackel and Cobb (1996) advanced the concept of social norms and proposed the construct of sociomathematical norms, suggesting that some mathematics-specific norms exist in mathematics classrooms, including, for example, what is considered mathematically sophisticated or an acceptable mathematics answer. In a recent study, Zhang et al (2022) explored Chinese and Australian students’ negotiative foci in terms of either facts and procedures, didactical norms, or social/interpersonal consideration, and found that compared with the Australian counterparts (8.95%), the Chinese pairs (4.84%) relatively less focus on negotiating didactical norms.

Since discourse is affected by the mathematics classroom’s cultural environment, different discourse environments can result in different mathematics learning. As students follow and shape certain social norms in classroom dialogue, how and to what extent can these discourse characteristics be embodied in collaborative mathematical problem-solving contexts? In collaborative mathematics problem-solving, students gradually achieve the group’s goal through communication and negotiation in the problem-solving process and complete or fail to complete the task. Therefore, examining the group’s negotiation process is not only an in-depth excavation of its members’ participation process but also a reflection of the cultural factors affecting it. Therefore, we examined five groups’ negotiation processes in collaborative mathematics problem-solving to understand the characteristics of group interaction, further analyse group members’ participation in tasks, and identify implications for future research and teaching practice. Specifically, this paper:

  1. (1)

    analyses the negotiative event chain for all discussion groups’ task completion processes, allowing researchers to more accurately understand and describe each group’s task completion process; and

  2. (2)

    codes and classifies each group’s task completion process to analyse and explore each member’s participation process in the different task completion processes.

The above exploration provides a foundation for understanding student participation in collaborative mathematics problem-solving, which may help students improve the quality of collaborative group discussion and teachers’ instruction in practice.

3.3 Materials and Methods

3.3.1 Data Source

The data in this study were derived from the Australian ARC Project: Social Essentials of Learning. This study focused on the collaborative problem-solving process of five groups in a class (Class 1B) of a middle school (LHZX) in City B taught by math teacher Zhang (pseudonym). There are two main reasons for recruiting these five groups, first, the five groups in this class were taught by the same teacher, so we assume that certain characteristics in terms of classroom norms could be determined. Second, the five groups have different gender composition in terms of number of female students and male students, which allow us to observe how gender composition may influence students’ collaborative work. In addition to student video data and classroom materials, after-school interviews with teachers were important in this study. During the interviews, teachers evaluated and explained group members’ behaviours, which was a source for understanding students’ participation behaviours. All five groups in this study completed the task to a certain extent, but each group’s problem-solving process and work differed. The group works as a whole completed the basic requirements of the task within an acceptable range. The commonality among these groups was that each had relatively abundant verbal interactions in various forms and relatively clear problem-solving paths in the group problem-solving process. Each group included four students and worked on the following mathematical Task one (see Appendix).

3.3.2 Analytical Approach

Based on the above purposes, this study analyses the video data and the transcripts of five groups that collaboratively work on a mathematical task. First, through repetitive observation of the recorded video, the researchers get familiar with the groups as well as the context of their collaborative work. On this basis, the transcriptions are used as another primary data. The researchers divide the transcribed text into negotiative event, determine the discussion topics in each negotiative event according to the discussion content of the students in the group discussion process, and further arranges all the topics as a chain which describes the problem-solving process of each group.

After obtaining the negotiative event chain, based on the similarity between different topics or the similarities of the functions of the negotiative events, the problem-solving process’ corresponding characteristics will be obtained. Combined with the recorded videos, the analysis of student participation during the problem-solving process will be given in terms of how students participate in each and a series of negotiative events.

Operationally, the following concepts need to be clarified.

  1. (1)

    Negotiative Event (NE): An NE is defined in this study as “an utterance sequence constituting a social interaction with a single identifiable purpose” (Chan and Clarke, 2017). The group discussion topic is the “single identifiable purpose” in this study.

  2. (2)

    Discourses: Discourses in this chapter refer to students’ complete or incomplete sentences spoken to themselves and each other during the discussion, as transcribed from the video. In the transcript, each sentence transition counts as a sentence, regardless of whether the sentence constitutes a full and strict sentence in literature.

  3. (3)

    Discourse volume: Discourse volume usually refers to the number of discourse sentences, as described above.

3.3.3 Validity

The validity of this study in defining negotiative events and dividing negotiative events mainly comes from the following guarantees. First, the definitions are clear and standardised. Clarke and Chan have deepened and consolidated the definition of negotiative events in the related literature and tested its validity in parallel Australian data (Chan and Clarke, 2008, 2019). Second, the coding process was examined by experts in education or mathematics education from our and other countries. After the definition and code were developed, they were reported to the International Classroom Teaching Research Center and the Mathematical Science Research Group of the University of Melbourne, Australia, respectively, until all experts recognised the final results.

In this study, the negotiative events in each group were coded by multiple people to determine their reliability. The initial coding time for all groups was from February 2019 to April 2019. In October 2019, the researchers re-coded and proofread all coding processes, with 94.7% consistency. Three groups were randomly selected for a third encoding in December 2019, reaching a 93.0% consistency. The error was controlled within five sentences. In the final stage, the researchers selected three groups to code independently with the other two researchers, discussed and negotiated uncertain divisions, and ensured that all three parties’ coding results were close to the same.

3.4 Results

3.4.1 An Overview of a Negotiative Event Chain

Tracing the topics discussed during the task completion process can determine each group's task completion process. The following Table 3.1 lists the negotiative event chain of each group. These negotiative event chain topics could describe how each group worked on the task and solved the mathematical problem.

Table 3.1 Negotiative event chain of the five groups
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Based on Table 3.1, two main characteristics of the five groups in the negotiative event chain during collaborative problem-solving can be obtained.

  1. (1)

    The number of negotiative events in each group varies

The above table shows that the number of negotiative events discussed by the five groups is different; 21, 21, and 20 negotiative events were determined by three groups of 1BG1, 1BG2, and 1BG4, respectively, a relatively similar amount, while the other two groups in 1B, 1BG3 and 1BG5, discussed 12 and nine topics, respectively. The number of negotiative events corresponds to the amount of discourse during the five group dialogues. Analysis of the discourse volume in the five group discussions shows that although the prescribed time to complete the task was the same, the discourse volume of each group in a given time differed. The Table 3.2 gives the five groups’ total utterances during the discussion; that is, the total number of sentences for all students.

Table 3.2 The total discourse volume of five groups
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According to this table, the groups’ discourse volume seems to correspond to the number of negotiative events determined. For example, groups 1BG3 and 1BG5 were very close regarding their number of negotiated events and total volume of speech, as were groups 1BG1, 1BG2, and 1BG4. We would wonder whether groups with similar numbers of negotiative events and discourse volume have similar characteristics in the problem-solving process. The next sections will provide a more in-depth analysis of the negotiation process.

  1. (2)

    The content involved is different—the intersection of pure mathematics and contextual knowledge

Based on the content knowledge involved in each negotiative event, the researcher clarified three types of NEs, including mathematics and task-related (MT) NE, non-mathematics but task-related (NMT) NE, and off-task (OFF) NE. The first type means that during this NE, students’ discussion focused on task-related mathematical knowledge, the second type means that students’ discussion was about the task context and certain related mathematical knowledge, and the third type NE means students’ discussion was off-task during these NEs. The following Table 3.3 summarises the distribution of different content negotiative events in each group during the entire discussion process.

Table 3.3 The number of different NE categories
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It can be seen from the above table that almost all group discussions were task-related. The proportions of MT and NMT events in groups 1BG1 and 1BG3 differed. Both groups showed relatively more discussions about non-mathematics and task-related topics and relatively fewer pure mathematical topics. The proportion of MT and NMT events was similar throughout the discussion for the other three groups.

3.4.2 Participation of Group Members in the Process of Task Completion

The above provides a brief overview of how the groups complete the tasks, but the details in terms of how each group member participates in the negotiation process and how the students interact with each other are still not clear. Combined with transcript analysis, the following sections will provide a more detailed analysis of the negotiation process from multiple perspectives.

3.4.2.1 The Effect of Discussion Content on Student Participation

Analysing the negotiative event chain deepens our understanding of the negotiation content involved in the task completion process. In the group task completion process, the negotiation content involved has a non-negligible effect on students’ participation in problem-solving. The number of MT and NMT events discussed by different groups differed throughout the discussion process, as did each member’s participation process.

Among the five groups in class 1B, the proportions of MT and NMT events in the 1BG1 and 1BG3 groups were quite different, with more NMT topics than MT topics tending to be discussed. Groups 1BG1 and 1BG3 were composed of three boys and one girl, and four boys, respectively; that is, boys accounted for a larger proportion of each group. As seen from the transcript and video, it seems that group members rarely kept raising questions about the mathematical content during the group discussion process. Individual students in the group undertook the entire mathematical calculation process and mathematical problem-solving. The following Table 3.4 shows the task completion of the two groups, 1BG1 and 1BG3.

Table 3.4 Worksheet of groups 1BG1 and 1BG3
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In fact, during the 1BG1 group’s discussion, S1 took the main role in completing the task, including drawing the figure, doing all the mathematical calculations, and so on; some corresponding data mismatches appeared in the apartment map as drawn (e.g., inconsistent width and length on both sides of the balcony, etc.). 1BG3’s situation was slightly different. While the mathematical data in the 1BG3 team’s task list was correct and met the requirements, almost all mathematical calculation tasks were undertaken and completed by S3 during the group’s discussion. Although other students questioned S3’s room design, layout, and other issues, S3 did not change the data and design and insisted on completing the task list according to his ideas.

On the other hand, for groups 1BG2, 1BG4, and 1BG5, the number of events for the two types of content was similar, with the discussion of MT and NMT events showing a certain regularity. For example, in the 1BG2 group, during the whole discussion process, MT events usually appeared in a series; that is, once the discussion of one MT event was initiated, the probability of the group continuing to discuss several MT events would be relatively high. The NMT events were similar, showing the phenomenon of getting together; that is, several associated events may be NMT events. A similar situation was also seen in 1BG5.

Undeniably, students must master mathematical knowledge and certain contextual knowledge about the task in the collaborative mathematics problem-solving process. The occurrence of a series of MT events had certain benefits for students to discuss and share mathematical knowledge fully. In the discussion process, students continued to extend their knowledge and realise or clarify their lack of understanding through group communication and discussion. The following are excerpts from group 1BG2 during NE6. In this segment, after S3 asked how many (square) metres were in 360 square centimetres, S1 and S2 estimated it as the area of a classroom floor tile and then, through this, estimated whether the apartment’s individual rooms were reasonably sized. After this incident, a series of MT incidents assessing whether the size of the bathroom was appropriate appeared.

S3 (NE6.2): Wait, wait, how many square metres equal to 360 square centimetres?

S1 (NE6.3): That’s too small.

S2 (NE6.4): I use a scratch paper.

S4 (NE6.5): She’s talking about a brick.

S1 (NE6.6): That should be 3.6 square decimeters.

S3 (NE6.7): That is 3.6 square decimeters.

S1 (NE6.8): How is 60 times 60 to 360?

S3 (NE6.9): 3600. That is 36 square decimeters, and then, 0.36 square metres, not even one square metre, then you say 4 bricks.

S1 (NE6.10): Four bricks are bigger than one square metre.

S4 (NE6.11): Four bricks make one square metre? What is 0.36 times 4?

S2 (NE6.12): If it is 60 square centimetres, you need 6 cm across.

S4 (NE6.13): 4 blocks are 1.44 square metres.

Note: S1, S2, S3, and S4 represent the four students, respectively, while NE6 is the event number. All the sentences in this event are coded in order; for example, the first sentence is NE6.1, and the paragraph starts from NE6.2.

As seen from students’ discussions, in the course of negotiation, whether the MT event string could be realised or caused depended on whether there was a group member who could ask and insist on the mathematical problem, such as, “how many square metres equal to 360 square centimetres?” In the above segment, S3 raised this issue for the second time in 1BG2’s group discussion. When first raised, the other group members did not answer; when asked again, students started to provide feedback and calculations and finally decided on an area standard that could be used for further calculations. A similar problem also occurred in the 1BG3 group. When S4 raised the question, “how big is the classroom” during the discussion process, it was ignored by other group members; after S1 responded, “I don’t know,” S4 did not insist on re-proposing. The reason for S4’s question in group 1BG3 should be the same as the reason for S3’s question in 1BG2 (i.e., corresponding to the actual area size rather than the mathematical area size); however, during the group discussion, due to various members’ participation, the interactive and dynamic nature of a certain problem cannot guarantee an immediate response or solution once a problem is raised. Therefore, the same question asked by different group members in different groups can also create different situations. Repeatedly asking the same questions, as S3 did in the 1BG2 group, can cause a string of MT topics.

3.4.2.2 Students’ Volume of Discourses Are Different

Analysing the negotiative event chain provides a basis for exploring students’ participation process in collaborative mathematics problem-solving and opens up a direction for analysing group members’ discourse volume in the whole negotiation process. In group problem-solving, each group member’s discourse volume can reflect their verbal participation in the negotiation process to a certain extent. By counting each student’s utterances throughout the entire discussion and calculating the proportion of all utterances, we can see how many words each student contributed to their group’s discussion.

Figure 3.1 shows the students’ percentage of speech volume in the five groups during the discussion process. The number of speeches (utterances) a group member makes during the entire negotiation process is their volume of speech and reflects their discourse contribution to the problem-solving process, to a certain extent.

Fig. 3.1
figure 1

The proportion of utterance amount in five groups

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In the 1BG1 group, the discourse volumes for S1 and S2 were relatively large, accounting for 36% and 31% of all utterances, respectively, while S3’s and S4’s discourse volumes accounted for only 14% and 18%, respectively.

In the 1BG2 group, S3 had the highest speech volume (38%), while S1 had the lowest (14%). S4 and S2 accounted for roughly the same amount of discourse, 23% and 25%, respectively.

In the 1BG3 group, S1’s and S3’s proportions of speech volume were similar (29% and 27%), with S4 contributing 25% and S3 19%.

In the 1BG4 group, S4, S3, and S2 contributed 30%, 27%, and 25%, respectively, while S1’s speech volume accounted for 18% of all utterances.

In the 1BG5 group, S1 spoke the least (6%), with S2, S3, and S4 accounting for 21%, 33%, and 39%, respectively.

To summarise, in the five groups of class 1B, except for the large difference in speech volume between S1 and other members in group 1BG5, the group members’ speech volumes were not very different. Comparing each group member’s discourse volume shows there were usually some members with more discourse volume and some others with less.

3.4.2.3 Students Have Different Levels of Control Over Tasks

Analysing the participation of group members from the perspective of topic content and discourse volume alone is not enough to describe the participation process because even if some members speak a lot or participate in specific topics, the participation process differs in other problem-solving stages. Further understanding group members’ control power in the group problem-solving process helps to describe individual students’ participation processes.

In the actual problem-solving process, the contextual task information that changes the discourse volume in each negotiative event differs in each group. For example, if the negotiative event with the most speech groups is excluded from the 20 negotiative events in the 1BG4 group (NE19; 39 utterances), the other events remain relatively stable. However, during the NE19 discussion, panellists experienced a relatively fraught row over whether the locations of the bathroom and study could be interchanged, which could reflect the relationship between group members to a certain extent.

The 1BG4 group comprises three girls (S1, S2, S3) and one boy (S4). In the NE19 discussion, G4S2, G4S3, and G4S4 were asked whether the positions of the bathroom and study room in the already formed draft need to be interchanged. Disputes arose, with S2 and S3 insisting they needed to be swapped and S4 disagreeing.

Symbol description: S1, S2, S3, and S4 represent the four students. NE19 is the event number. All event sentences are coded in sequence, e.g., the first sentence is NE19.1.

S2 (NE19.5): You draw, you draw. Why is this set of data so messy? What is 6 square metres? 6 square metres is a balcony, what is 5 square metres?

S3 (NE19.6): Toilet.

S4 (NE19.7): 5 square metres?

S2 (NE19.8): 5 square bathroom?

S3 (NE19.9): Can’t we just change the balcony and bathroom?

S1 (NE19.10): No time.

S3 (NE19.11): No problem, just change its label.

S4 (NE19.12): Oh, you don’t have to change it, you can just do it.

S3 (NE19.13): Have you ever seen a balcony bigger than a bathroom?

S2 (NE19.14): The bathroom is smaller than the balcony.

S4 (NE19.15): Huh?

S2 (NE19.16): The bathroom should be bigger than the balcony, otherwise how can you stay there?

S4 (NE19.17): My bathroom is smaller than the balcony.

S2 (NE19.18): My grandma’s bathroom is too big.

S4 (NE19.19): That’s it, no time.

S3 (NE19.20): Changed the label——

S2 (NE19.21): Change the label and the words.

S3 (NE19.22): The location does not need to be changed.

S4 (NE19.23): But you also changed the size…

S3 (NE19.24): Isn’t that the end of the name change?

S4 (NE19.25): that data…

S3 (NE19.26): Our data is on scratch paper.

S4 (NE19.27): You should change if you like.

S3 (NE19.28): Changed the name, but the data remains the same.

S4 (NE19.29): Well, change if you like.

S3 (NE19.30): The balcony is still small, how can I get it?

S4 (NE19.31): Change it if you like.

S2 (NE19.32): Yes, then we will change it.

S4 (NE19.33): What else to say.

S2 (NE19.34): Yes, he was nagging there, nagging like an old woman.

…

(This paragraph shows that the disputing parties did not have a common understanding and did not seek the others’ understanding.)

The above intercept shows the quarrel between the members of the 1BG4 group during the final stages of the group discussion. S2 first raised a question about matching the relevant data between the room and the room area, causing the rest of the group to address and clarify the size and function of the various rooms. S4 responded to S2’s question and proposed that five square metres had been designated for the bathroom. S4 and S2 then tried to verify whether a five-square-metre bathroom was appropriate. S3 thought that the bathroom was too small and proposed changing the positions of the bathroom and balcony. At this point, S1 expressed concerns about the time to complete the task. S3 again proposed simply swapping the names of the two rooms on the task list. S4 also raised objections and questions at this time. To this point, both parties’ intentions were relatively clear. S4 thought the bathroom should not be replaced, while S2 and S3 insisted on changing it. S1 raised concerns about time but did not explicitly oppose or approve the swap.

In the ensuing discussion, the two sides gave reasons for the exchange. S2 and S3 believed the bathroom should not be smaller than the balcony, which S4 did not recognise. S2 and S3 proposed changing the rooms’ labels, stating there was no time to redraw the plans. S4 believed that when the rooms were relabelled, the data would also change accordingly, so changing the label was not just changing the name. S2 and S3 then proposed that the data were draft data and could be changed. In the end, S4 compromised and ended the dispute with, “change it if you like.”

Analysis of the above dialogue shows that the process that caused and evolved the dispute was as follows. First, before the discussion began, the two parties had inconsistent understandings of whether the bathroom or balcony was larger. S2 and S3 believed the bathroom should be larger than the balcony, while S4 believed it could be smaller. During the discussion, although S2 and S3 suggested that it would be fine just to change the name, S4 believed that would lead to a change in the data (and thus necessitate recalculation). At this point, S2 and S3 might not have understood S4’s thinking nor responded to S4’s questions from the perspectives of data and mathematics, and the dispute reached its peak. In the end, S4 “reluctantly” accepted the proposals of S2 and S3, and S1 changed the label in the graphics. S4 used “you” several times in the final discussion to refer to and differentiate themself from S2 and S3, reflecting the separation of the whole and its parts.

Although the reason for the dispute started from a lack of consensus, neither party carefully considered and calculated the other’s proposals during the discussion, so it was difficult to form a consensus. S4’s grudging compromise and separation confirm the lack of true consensus and, to a certain extent, reflect that S4 was on the “weak” side of the group discussion.

Figure 3.2 illustrates the evolution of the group discussion’s content and the group members’ participation in discussing NE18, dividing it into three stages. Based on this process, the group discussion was affected by establishing consensus among subjects, the relationship between the whole and the individual, and (because S4 was the sole boy) possible gender differences. While these factors were reflected in a single negotiative event, they may exist in all of them. The negotiative event chain can also indicate the influence of different factors, which may intensify during the discussion. Therefore, team members’ engagement throughout the task completion is closely related to the procedural changes in consultation on specific issues.

Fig. 3.2
figure 2

The evolution model of NE18

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In the above discussion process, S2, S3, and S1 had relatively stronger control over the task than S4. Therefore, both from the negotiation result (the group did not accept S4’s proposal) and the negotiation process (the group did not accept S4’s proposal), a temporary intra-group separation occurs, and S4’s control over the negotiation process continues to weaken, making their participation change.

The same phenomenon appeared in other groups, reflecting a gradual transfer of control over the problem-solving process. In the 1BG2 group, for example, although all group members initiated the task completion process, control gradually transferred to individual members as the discussion progressed. Since the negotiative event S2 initiated did not match the current task content, their participation in and control over the process gradually declined, weakening the entire participation process.

3.5 Summary

This chapter achieves two research goals. First, providing an overview of the negotiative event chain facilitates understanding the task completion process and how students negotiate on different topics to complete a task in each group. Second, based on the negotiative event chain, students’ participation and negotiation details are interpreted from different perspectives regarding the discussion content, students’ utterances amount, and group status in terms of collaboration or disparity in negotiation.

Corresponding to the problems and tasks in this study, the five groups reflected specific and individual characteristics. First, although the number of negotiative events related to the negotiative event chain differed in each group, most were related to the mathematical problem. Second, the amount of discourse about negotiative events in each group’s negotiative event chain varied. Third, the content of negotiative events shows there were more MT events than NMT events, i.e., the students discussed mathematical content less than contextual content. Individual groups or classes also exhibited characteristics that provided new perspectives for further understanding group participation in problem-solving.

In addition to analysing each group’s problem-solving process, the group members’ participation process was explored from three perspectives—discussion content, student utterances, and group status in negotiation—with the first two providing a basis for exploring the last. The study found that (1) specific negotiation content probably impacted group members’ participation in the task completion process; (2) group members’ discourse volume proportion showed a degree of gender difference in each group; and (3) group members’ responses to a particular sub-task may influence their participation in the overall task completion process.

The research in this study further consolidates the arguments already made in the previous research; that is, student participation in collaborative mathematics problem-solving needs to be investigated from various perspectives. The study provides certain empirical trials in terms of understanding how Chinese students work collaboratively in mathematical tasks. As seen from current literature, although the east–west comparison about mathematics teaching and learning has been widely conducted, it rarely reveals the characteristics of student interactions among Asian students. Most researchers who state that collaborative learning can be beneficial for students’ mathematics learning are based on research conducted in Western culture (Xu and Clarke, 2019). To investigate how collaboration may happen in Asian student groups and whether such collaborative work contributes to students learning, it is essential to first understand how students interacted with each other in groups. Therefore, such trials can pave the way for readers to understand the negotiation process in each group’s task completion process.

However, the analysis in current study is still at a macro level; more combined micro- and macro-level analyses are needed in future studies. On the one hand, more in-depth analysis about student interaction and their mathematics learning in Asian classrooms should be done so as to verify how collaborative learning actually plays its role in both students’ mathematics learning and Asian mathematics classrooms. On the other, if that collaborative learning does contribute to students’ learning, how should teachers and students make efforts on facilitating this benefits through teaching and learning, how can educators help teachers and students in practice should be further investigated.