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.

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.

(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.

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),

**Fig. 3.2** The evolution model of NE18

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.

## **References**

Brousseau, G. (1986). Fondements et méthodes de la didactique des mathématiques. *Recherches en Didactique des Mathématiques, 7*(2), 33–115. http://rdm.penseesauvage.com/Fondementset-methodes-de-la.html

Brousseau, G. (2002). *Theory of didactical situations in mathematics: Didactique des mathéatiques, 1970–1990*. Kluwer Academic Publishers.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Chapter 4 Research on Individual Authority and Group Authority Relations in Collaborative Problem Solving in Middle School Mathematics**

## **4.1 Introduction**

Classroom teaching practices have attracted the increasing attention of researchers (Li et al., 2022; Zhao et al., 2022). In teaching practice, front-line teachers are committed to promoting class management based on a fair and equitable system or the theory of teacher-centred authority in class management. Researchers have focused on examining teacher authority while paying insufficient attention to student authority. Teachers should not only pass knowledge to their students but also encourage them to think and learn by themselves, which is good for teaching and learning (Díez-Palomar et al., 2021). Good teacher-student relationships come from the integration of teacher authority and student authority. However, very little attention is paid to student authority in mathematics education. Collaborative problem solving (CPS) can be a good vehicle for exploring student authority. It provides a mathematical learning environment that involves non-teacher-led activities. In CPS, students use a range of mathematical and non-mathematical forms of language to gain authority, which influences the process and outcome of learning. For these reasons, the study of individual authority and group authority relations in CPS in mathematics is bound to become particularly important.

J. Wu (B)

Changjun Banmahu School, Changsha, China e-mail<EMAIL_ADDRESS>
<sup>©</sup> The Author(s) 2024

Y. Cao (ed.), *Students' Collaborative Problem Solving in Mathematics Classrooms*, Perspectives on Rethinking and Reforming Education, https://doi.org/10.1007/978-981-99-7386-6\_4

## *4.1.1 Conceptual Framework*

#### **<IP_ADDRESS> Authority**

The term "authority" is widely used in social life and has different meanings in different cultures and contexts. In the Chinese dictionary (*Ci Hai*), it refers to two concepts: (1) power and prestige, and (2) a force of prestige and dominance developed in the course of human society. Also, there are different interpretations of authority in different disciplines. From a sociological perspective, authority is a force that convinces people without pure violence based on two elements: voluntary obedience and belief. In psychology, there are two manifestations of authority: formal and informal, which highlight the influence individuals and groups have on other people and groups.

There are also different interpretations of authority. Despite different disciplinary perspectives, all definitions reflect at least two characteristics. First, authority emphasises the relationship of obedience between authority objects and other authority subjects, based on value recognition. Second, the operation of authority produces a relationship between authority objects and other authority subjects in terms of influencing and being influenced. It is the recognition of authority objects to authority subjects in terms of ideology and obedience in terms of behaviour.

This study examines changes in the relationship between individual authority and group authority relations in CPS in mathematics. The definition of student authority used herein is based on the theory that authority emphasises the obedience of authority objects to authority subjects. This authority relationship between three and more people is the theoretical basis for the definition of group authority relations. Students' group authority relations change through interactions, and changes in authority are dynamic.

#### **<IP_ADDRESS> Student Authority in Mathematics Activities**

This study examines the relationship between students' individual and group authority in CPS. In teacher-empowered student collaboration and management of student-led CPS activities, authority shifts from being unilaterally held by the teacher to being distributed between the teacher and the students. Mathematics activities research has begun to examine the authority relations between students.

In terms of research on authority in classrooms, several studies have pointed to definitions of authority in classrooms. Cohen (1994) defined authority as "an agreed-on rank order where it is generally felt to be better to be high than low rank." Status can be thought of as a relationship of power among peers. That power can be academic, as in status, derived from perceived smartness, or social, as in status derived from popularity. This somewhat vague positioning of authority highlights its intellectual and social categories. Ernest (2008) proposed that a teacher "has two overlapping roles—namely as director of the social organisation and interactions in the classroom (i.e., social controller) and as director of the mathematical tasks (i.e., task controller). This distinction corresponds to the traditional separation between being 'in authority' (social regulator) and being 'an authority' (knowledge expert)." Researchers defined authority as a resource of control associated with the right to lead and the obligation to follow (Amit & Fried, 2005; Ernest, 2008; Pace & Hemmings, 2007). Teachers can be both an intellectual authority based on their knowledge and a social authority based on their power to issue instructions to students and control their behaviour. Boaler and Greeno (2000) believed that social authority always operates in classrooms, occurring wherever humans interact. Intellectual authority relations are at play when individuals are engaged in intellectual work, defined in schools as engaging in academic tasks.

From the above scholars' explanations of authority in mathematics education, student authority relations in CPS in mathematics can be divided into social authority and intellectual authority. The ACT21S project "CPS" consists of two main components, "Collaboration" and "Problem solving." The problem solving component consists mainly of skills required to solve problems. Intellectual skills reflect the management of tasks and include task management, learning, and knowledge building. This paper combines this perspective to define individual authority and group authority relations in CPS.

#### **<IP_ADDRESS> Student's Individual Authority in CPS in Mathematics**

The definition of individual authority is derived from Langer-Osuna (2016). Students' individual authority refers to the students' personal intellectual and social authority in CPS activities. Students' individual intellectual authority refers to the fact that students' individual behaviour is a useful source of information (or lacks such credibility). A student's individual social (directive) authority refers to the student being deemed to have (or not have) the right to issue directives to group members.

#### **<IP_ADDRESS> Group Authority Relations in CPS in Mathematics**

The definition of group authority is derived from Langer-Osuna et al. (2020b). Group authority relations are formed as a result of the authority of three or more people operating in the group. Group authority relations refer to an intellectual or social relation of submission between authority subjects and authority objects, which is also a relationship of influencing and being influenced. In the current research, seven types of group authority relations can be formed in cooperative groups as intellectual and social authority compete and disperse. Group authority relations are reflected in students' utterances and behaviours in classroom communication and are dynamic in their changes.

#### **<IP_ADDRESS> Conceptual Framework for Individual and Group Collective Authority Relations**

This section defines students' individual authority and group authority relations in CPS in mathematics. This diagram explains how individual authority forms group authority relations and the types of group authority relations.

Figure 4.1 explains the relationship between individual authority and group collective authority. (1) Individual authority is composed of intellectual and social authority. In a group of three or more, individual authority forms group authority relations; specifically, individual intellectual authority forms group intellectual authority relations and individual social authority group collective social authority relations. (2) Group intellectual authority relations include shared and concentrated intellectual authority relations; contested intellectual authority relations are formed through students' sharing of, concentration of, or competition for intellectual authority. (3) Group social authority relations include shared, concentrated, and contested social authority relations, as well as disbanded social authority relations due to the sharing, concentration, competition, or dissolution of students' social authority. (4) These seven different group authority relations have conceptual crossovers. Shared intellectual and shared social authority relations belong to shared authority relations. Concentrated intellectual and concentrated social authority relations belong to concentrated authority relations. Contested intellectual and contested social authority relations belong to contested authority relations.

**Fig. 4.1** Conceptual framework for individual authority and group authority relations

#### **<IP_ADDRESS> Research Questions**

Through exploring the characteristics of authority within different structured groups, it is possible to explore the potential relationship between authority and performance. Further, the transformation of authority possibly influences students' activity during CPS (Langer-Osuna et al., 2020b). Based on a video analysis of CPS activities in a middle school, this study examined student authority relations in the CPS stage to answer two research sub-questions:

Question 1: What is students' individual authority in high- and low-scoring groups in CPS in middle school mathematics?

As this study explores individual and group authority relations, it first explores individual authority (individual intellectual authority and social authority) in high- and low-scoring groups in CPS in mathematics.

Question 2: How do students distribute and shift group authority relations in highand low-scoring groups in CPS in middle school mathematics?

Individual authority interacts in groups of three or more to form group collective group authority relations. This question explores the specific distribution and variation of the seven different group authority relations.

## **4.2 Research Design**

This study focused on the students' authority relations in CPS in middle school mathematics. Two sub-problems were used to explain the characteristics of individual and group authority in CPS. The overall idea of this research is shown in Fig. 4.2.

This research mentality diagram depicts the concepts: of (1) node (related group authority relations proposal negotiation unit); (2) coverage rate (the ratio of node dialogues generating group authority relations to all dialogues); and (3) individual authority rise and fall (the frequency with which an individual's proposal is accepted or rejected).

## *4.2.1 Data Sources*

Purposeful sampling is applied for the current research. Six four-student groups were selected for the study. The six groups were drawn from Teacher A's classes (C01a and C01b) and Teacher B's classes (C02a and C02b) (see Table 4.1) within the same school. This study further divided the six groups into three high-scoring and three low-scoring groups for more pertinent analysis, based on the group score table for mathematical collaboration problem solving (Appendix 4.1). Students scoring above 6 were in the high-scoring group; students scoring below 6 were in the low-scoring

**Fig. 4.2** The overall research roadmap of this study

**Table 4.1** The attributes of the object group in this study

group. The three high-scoring groups received scores of 10, 9, and 7, respectively, while the three low-scoring groups received scores of 5, 5, and 3. The groups selected were those who spoke up most actively in classes.

## *4.2.2 Data Analysis*

This study chose a qualitative research method based on video recordings. After transcribing the video dialogues, video text analysis methods were used to generate statistics on the rise and fall of individual students' authority in the high- and lowscoring groups. Student authority relations in CPS in mathematical activities were explored by counting group authority relation nodes and different group authority relations as a percentage of coverage (as a ratio of total discourse). This study was conducted using NVivo version 12.

## *4.2.3 Coding Scheme*

This sub-section describes two research tools used to explore the individual authority and group authority relations in CPS in mathematics. There were several reasons for selecting and adapting these two research instruments. The first reason was the similarity of the two research samples. The original scales were initially used to study 10–11-year-old students' CPS in mathematics activities; this study involved Grade 7 students' CPS in mathematics activities. The second reason was the operability of the research codes. The research tools provide operational definitions that can be analysed in classroom videos, making both individual and group authority relations more visible. Third, both research tools have a strong theoretical basis, making them persuasive.

(1) An analytical tool for coding individual authority in CPS in middle school mathematics

This study drew on Langer-Osuna's (2016) findings on individual authority, adapting them accordingly to form an individual authority coding analysis tool in conjunction with this study (see Table 4.2). This study omitted the coding of individual authority statistics related to teachers assessing the quality of students' arguments and the merits of students' behaviour. The groups consisted of four people, two boys and two girls, coded as B1, B2, G1, and G2.

This study was conducted to represent the dynamics of students' individual authority and authoritative characteristics through their individual intellectual and social authority statistics. Individual authority rise and fall statistics refer to the frequency of each group member's successful and devalued bids for intellectual authority and social authority. Engle et al. (2014), in their work, *Toward a model of influence in persuasive discussions: Negotiating quality, authority, privilege, and access within a student-led argument*, referred to a proposal negotiation unit (PNU). A PNU is a set of interactions that begin with a discourse that makes a suggestion around a problem (for example, presenting an idea to be evaluated or giving an instruction). The data for this study were analysed at the event level of the PNU, where a PNU is a group authority relation node. Table 4.3 explains the methodology

**Table 4.2** Coding of individual authority statistics for CPS in mathematics in this study

for the rise and fall statistics of individual authority in a given PNU. For example, a suggestion might include, "I know what to do, let's add numbers." This would be coded as a bid for intellectual authority; a group response such as "yes" after adding the numbers would be coded as a successful acceptance of the bid, positioning the first speaker's intellectual authority. Conversely, a response such as "no" would be coded as a rejection of the bid, thereby devaluing the first speaker's perceived authority. In this study, B1 represents the first male, G1 the first female, and so on.

Figure 4.3 shows the results of CPS in mathematics for the groups corresponding to the authority relation nodes in Tables 4.2 and 4.3. Students' individual authority statistics were based on students' conversations and the results of CPS in mathematics.

Students' interactions were qualitatively analysed after coding. The interplay of absorption frequency (represented by positive signs) and rejection frequency (represented by negative signs) were analysed through a specific CPS in mathematical videos.

**Table 4.3** Schematic table of the way individual authority statistics for CPS in mathematics are presented

**Fig. 4.3** Apartment layout task sheet in Table 4.3

(1) Tools for analysing group collective authority coding in CPS in middle school mathematics

Based on Langer-Osuna's (2020a) coding of group collaborative authority relations in CPS in mathematics, this study developed the group authority relations coding analysis tool. The individual authority mentioned above forms group authority relations in groups of three or more. Students' individual intellectual authority forms intellectual authority relations in groups of three or more. The group's intellectual and social authority can be transformed between shared authority, concentrated authority, contested authority, and disbanded authority. There are seven different group authority relations. This study examined the different social and intellectual authority relation nodes and coverage and their translation between collaborative group collectives. Table 4.4 shows an adaptation of Langer-Osuna's (2020a) coding of group authority relations to form an analysis tool for this study. The study was interpreted in the context of a CPS task in middle school mathematics called "Xiao Ming's Apartment".

This study first set intellectual and social authority relations as the primary codes. Shared, contested, and concentrated intellectual authority relations; shared, contested, and concentrated social authority relations; and disbanded social authority relations were set as secondary codes. Conceptual crossover occurred between the seven different group authority relations. Shared intellectual and social authority relations belong to shared authority relations, concentrated intellectual and social authority relations belong to concentrated authority relations, and contested intellectual and social authority relations belong to contested authority relations.

The data in this research were analysed at the PNU level. A PNU is a group authority relation node. For details, see the example in Table 4.3. This study counted the variation in social and intellectual authority relations across different groups and the nodes and continuity accounted for different types in the 15 min CPS video.

## **4.3 Results**

## *4.3.1 Individual Authority Study Results of the Highand Low-Scoring Groups*

This section explores statistics on the rise and fall of students' individual authority in CPS, referring to the frequency of each group member's successful and devalued bids for intellectual and social authority. B1 and G1 represent the first boy and first girl in the group, and so on. Specific statistical methods are explained in Table 4.3. For example, B1's suggestion, "I know what to do, let's add numbers," would be coded as a bid for intellectual authority; positive group responses (e.g., "Yes") after adding the numbers would be coded as a successful acceptance of the bid and recorded as B1 (+1), locating the first speaker's knowledge authority. Conversely, negative responses (e.g., "No") would be coded as a rejection of the bid and recorded as B1

**Table 4.4** Coding of collective group authority relations in CPS in mathematics in this study

(−1). The frequency of absorption (a positive sign) and rejection (a negative sign) are shown in Table 4.5.

In high-scoring group 01a-02, B1 had intellectual authority bids accepted six times, while G1 had intellectual authority bids accepted eight times. While the difference in intellectual authority between the two was not significant, their authority bids were accepted significantly more often than those of the other two students in their group. In groups 01b-02, B2's intellectual authority bid was accepted seven times and G2's social authority bid was accepted six times. Group 02a-04's B2 and G2's intellectual and social authority bids were also accepted more often than the

**Table4.5**Statisticsontheriseandfallofindividualauthorityinthehighandlow-scoringgroups other students in this group. Therefore, for the high-scoring groups sampled, group members' authority bids were accepted more often, with one or two students having significantly higher individual intellectual or social authority than the other students in their group.

In the low-scoring group, the frequency of authority bids being accepted and rejected authority bids was more balanced across group members. For example, in group 02b-05, B4 made four intellectual authority bids, but they were rejected three times in favour of other group members' authority bids. Based on Tables 4.5 and 4.6 offers a statistical summation for the individual social and intellectual authority for the high- and low-scoring groups.

Table 4.6 shows that there were 111 authority bids in the high-scoring group and 98 in the low-scoring group, indicating that individuals in the high-scoring group were more inclined to contribute to CPS in mathematics. There were more intellectual authority bids than social authority bids in both the high-scoring (64 vs 45) and lowscoring (64 vs 34) groups. The high-scoring group had a higher intellectual authority summation (26) than the low-scoring group (10). It also had a significantly higher total authority summation (31) than the low-scoring group (12). Thus, it can be concluded that students in the high-scoring group were able to have their proposals endorsed more often than the low-scoring group.

## *4.3.2 Group Collective Authority Study Results for the Highand Low-Scoring Groups*

The first part analyses the intellectual and social authority relations in the highand low-scoring groups under primary coding. The second part offers a comparative analysis of the different authority relations in the high- and low-scoring groups under secondary coding.

#### **<IP_ADDRESS> Nodal Analysis of Intellectual Authority Relations and Social Authority Relations in High- and Low-Scoring Groups**

In this study, relevant authority relation nodes were counted and coded. Teacher intervention discourses were not included into coding. The coding process removed non-responsive self-monologues and conversations after members had stopped using their pencils for the task.

In this article, the layout of the apartment, dimensions of the rooms, and names of the rooms (kitchen, living room, bedroom) are in the category of intellectual authority relations. Specific functions (toilet shower, bedroom sleeping) belong to the category of social authority relations. The PNU mentioned above is a set of interactions that begins with a discourse making a proposal around a coded component (e.g., presenting an idea to be evaluated or giving an instruction). The table below

Statisticaltableofindividualauthorityresultsforhighandlow-scoringgroups

shows an example of a contested authority negotiation unit, where a PNU represents an authority relation node.

In the last authority relation event, it can be concluded that B1 and G1 created a relevant dispute over the size of the apartment and proposed separate solutions to the problem. Some were accepted and some were rejected. Events like these were defined as contested intellectual authority. Similarly, authority relation PNUs identified throughout the CPS in the mathematics process were noted as authority relation nodes (Table 4.7).

The high-scoring group had 38 authority relation nodes, compared to 57 for the low-scoring group. The high-scoring group had more intellectual authority relation nodes than social authority relation nodes, while the low-scoring group had the opposite. The difference in the number of intellectual and social authority relation nodes between the high- and low-scoring groups was not significant. Also, based on the specific video recording text analysis, this study found that in terms of intellectual authority relations, shared authority was the most frequent, and concentrated authority was the least.

**Table 4.7** Nodal table of intellectual authority relation and social authority relation for high- and low-scoring groups

**Table 4.8** Table of coverage of intellectual authority relation and social authority relations for high- and low-scoring groups

The volume of discourse was counted using NVivo application to determine related coverage, referring to the ratio of conversations corresponding to nodes of authority relations to total conversations. Table 4.8 counts the ratio of group social authority relations and social authority relations under the group authority relations level code.

The average authority relation coverage for the high-scoring group was 71.7%, roughly the same as for the low-scoring group (71.08%). Both the high- and lowscoring groups had more intellectual than social authority relations. In terms of average coverage, both groups had more intellectual than social authority relations. The high-scoring group's higher average intellectual authority coverage was more pronounced than its higher coverage of social authority relations.

#### **<IP_ADDRESS> Analysis of Specific Changes in Different Authority Relation Nodes in High- and Low-Scoring Groups**

Relevant authority relation nodes were counted in this study. The high-scoring group had a total of 38 authority relation nodes, of which 22 were intellectual authority relation nodes and 16 were social authority relation nodes. The low-scoring group had 57 authority relation nodes, of which 26 were intellectual authority relation nodes and 31 were social authority relation nodes. This section lists all authority relation nodes by names in chronological order. A line graph is used to illustrate the changes in their specific authority relations. The horizontal axis has 25 authority relation nodes. The vertical axis represents the seven authority relations: shared, contested, and concentrated intellectual authority relations; shared, contested, and concentrated social authority relations; and disbanded social authority relations. This diagram facilitates observation of the specific changes and continuity of different group authority relations, based on Table 4.7 (Figs. 4.4 and 4.5).

#### 4 Research on Individual Authority and Group Authority Relations … 91

**Fig. 4.4** Change in authority node for high-scoring group

**Fig. 4.5** Change in authority node for low-scoring group

From the previous section, it can be concluded that there were more intellectual than social authority relations in the high-scoring group. In terms of stability, the highscoring group had less variation in authority relation nodes. In both the high- and low-scoring groups, shared intellectual authority nodes were the most common and concentrated authority was the least common throughout the 15-min mathematical CPS task.

In summation, there were 12 shared intellectual authority nodes in the highscoring group and 17 in the low-scoring group. In terms of specific changes in group authority relations, the high-scoring group's 12 shared intellectual authority nodes produced contested and concentrated social authority relations without obvious tendency. However, the low-scoring group's shared intellectual authority relations were very likely to develop into disbanded social authority relations. In the 02b-05 group, there were five "shared intellectual authority relation → disbanded social authority relation" changes, compared to only two in the 01b-01 group with only seven authority relation changes. In terms of specific changes in their successive intellectual authority, contested authority and shared authority always alternated over time, with shared intellectual authority relations developing into contested intellectual authority relations and contested intellectual authority relations developing into shared intellectual and shared social authority relations. In terms of social authority relations, contested and disbanded social authority alternated. Contested social authority was prone to becoming disbanded social authority in the group.

#### **<IP_ADDRESS> Comparison of Shared, Contested, Concentrated, and Disbanded Authority Relations in the Highand Low-Scoring Groups**

This study proposes a crossover between primary and secondary codes in the conceptual framework for group authority relations. There was conceptual crossover in the seven different group authority relations (shared intellectual and social authority relations belong to shared authority relations; concentrated intellectual and social authority relations belong to intellectual authority relations; and contested intellectual and social authority relations belong to contested authority relations), thereby forming four different group authority relations: shared authority relations, contested authority relations, concentrated authority relations, and disbanded authority relations (Fig. 4.6).

Shared authority was the highest of all authority relations for both the highand low-scoring groups. Each group worked together to make intellectual or social contributions to solve the problem. Concentrated authority relation nodes were the least represented, accounting for only 4.22% of all nodes in the low-scoring group and reflecting that students shared authority during CPS in mathematics rather than clustering authority on the same person. The low-scoring group had a 23.27% rate of disbanded authority, second only to shared authority. The differences in the percentage of coverage of this component between the high- and low-scores groups were significant. Engaging in off-task activities significantly negatively influenced the low-scoring group's CPS outcomes.

**Fig. 4.6** Comparison of shared, contested, concentrated, and disbanded authority relations coverage for high- and low-scoring groups

#### **<IP_ADDRESS> Comparison of Shared, Contested, Concentrated, and Disbanded Authority Relations Under Intellectual and Social Authority in the High- and Low-Scoring Groups**

Figure 4.7 shows a comparison of authority relation coverage for the seven different group authority relations under Level 2 coding.

After secondary coding, intellectual authority relations represented a very high proportion of shared authority relations, 28.08% in the high-scoring group and 17.79% in the low-scoring group. Shared, focused, and contested intellectual authority relations were higher in the high-scoring group than in the low-scoring group. There was little difference between the high- and low-scoring groups in contested intellectual authority. The low-scoring group had the highest disbanded social authority coverage. Disbanded authority relations are not conducive to high scores in CPS.

## **4.4 Conclusion**

The study found that students in high-scoring groups had more individual authority bids and acceptances. The imbalances in individual authority predisposed to high scores in CPS outcomes.

**Fig. 4.7** Comparison of coverage of different authority relations under Level 2 coding for highand low-scoring groups

## *4.4.1 Analysis of Individual Authority in CPS in Middle School Mathematics*

#### **<IP_ADDRESS> More Individual Student Authority Bids and Acceptances in High-Scoring Groups Than in Low-Scoring Groups**

The total number of bids was greater for the high-scoring group members, indicating they were more likely to contribute to CPS in mathematics than their low-scoring peers. Members made more intellectual than social authority bids in both groups. The authority summation was significantly higher in the high-scoring group, indicating its members' proposals were approved more often than in the low-scoring group.

Students in high-scoring groups were more likely to express and address taskrelated ideas, accept criticism, and respect other group members' opinions. The results indicated that timely feedback is an important feature of deep discussion. Group members' ability to give reasoned explanations for others' questions or timely feedback on others' suggestions facilitated deeper group discussion.

The unidirectional and bidirectional connections arising between the four factors are presented in Engle et al. (2014) influence model. The more verbal or eye contact is made, the more students enter the conversational layer and interaction space, making it easier to produce high-quality arguments and thereby increase authority. This is a cyclical process that ultimately affects CPS outcomes. In conjunction with this study, it can be concluded that a bold approach to the articulation of mathematical reasoning ideas and an openness to criticism are factors that promote good CPS outcomes in middle school mathematics.

#### **<IP_ADDRESS> Imbalance in Individual Authority Makes It Easy for CPS to Result in High Scores**

Authority bids were accepted more often in the high-scoring group, with one or two students within each high-scoring group having significantly more individual intellectual or social authority than the other students. Authority bids were more balanced across group members in the low-scoring groups. For example, in the 02b-05 group, B4 made five intellectual authority bids but was rejected four times, while other group members made fewer authority bids. The low-scoring group was more balanced in terms of the frequency of authority bids being accepted and rejected. Other research on the role of social skills in middle school leadership in problem solving activities (Sun et al., 2017) indicates that groups that experience discussion achieve better problem solutions. In conjunction with the findings of this study, this shows that one to two leaders must be present in each group to lead other members through CPS and make it more effective.

## *4.4.2 Analysis of Group Authority Relations in CPS in Middle School Mathematics*

The characteristics of group authority relations in CPS in mathematics in middle school were derived by counting the group authority relation nodes (authority nodes) and the discourse coverage of different authority relations in students' groups throughout the CPS, and then classified and interpreted in this study.

#### **<IP_ADDRESS> Shared Intellectual Authority Relations Are Most Conducive to Producing CPS in Mathematics Results**

The specific variation in nodes showed that the number of shared intellectual authority nodes was highest across authority relations for both the high- and low-scoring groups. The frequency of intellectual authority relations was generally higher than for social authority relations in both the high- and low-scoring groups, so the group contributed intellectually or socially to problem solving. Video text analysis revealed that the shared intellectual authority relations period produced the most CPS results.

#### **<IP_ADDRESS> Least Occurrence of Concentrated Authority Relations in CPS in Middle School Mathematics**

The lowest percentage of concentrated authority was found in the high-scoring group, with 1.1% of concentrated social authority. No concentrated social authority relations nodes appeared in the low-scoring group, reflecting that students shared authority during CPS in mathematics and teachers dispersed student authority during CPS.

#### **<IP_ADDRESS> Contested Authority as a Catalyst for Authority Change in CPS in Mathematics**

Contested authority was second only to shared authority in both the high- and lowscoring groups, while contested intellectual authority was more predominant in terms of its secondary coding. This study has shown that contested and shared authority always alternated over time. Specifically, shared intellectual authority relations later developed into contested intellectual authority, while contested intellectual authority relations later developed into shared intellectual authority relations, facilitating CPS outcomes.

Cobb (1995) noted that argument for authority can be productive in the classroom, facilitating the more equitable distribution of authority and supporting different students' opportunities to learn. However, from an intellectual perspective, that can hinder the formation of CPS outcomes. Previous studies (Langer-Osuna, 2011; Langer-Osuna et al., 2020a) have shown that contested social authority may disrupt the process in ways that reshape the dynamics of cooperation. Taken together, the existing research and the present study's findings suggest that contested authority catalyses excellent CPS outcomes with contested intellectual authority and low CPS outcome scores with contested social authority under secondary coding.

#### **<IP_ADDRESS> More Stable Changes of Authority Relations Promote High Scores CPS in Mathematics Outcomes**

The high-scoring group had more intellectual authority relation nodes than social authority relation nodes, while the opposite was true in the low-scoring group. The high-scoring group also had more stable authority relation performance, with more coverage of social authority relations than the low-scoring group. Shared intellectual authority relations were more unstable in the low-scoring group. The "shared intellectual authority relation → disbanded social authority relation" process changed several times during the task solving process. Non-engagement with on-task activities occurred after shared cooperation. Stable shared intellectual authority relations were most conducive to producing CPS in mathematics outcomes. As Langer-Osuna et al. (2020a) pointed out, shifts in social authority relations are more dynamic than shifts in perceived authority. Combined with the characteristics of group authority relations in this study, the high-scoring group had a higher proportion of intellectual authority relations.

#### **<IP_ADDRESS> Summary**

The current research has revealed the characteristics of high/low-scoring groups. In the current research, more individual student authority bids and acceptances and more imbalance of authority within groups are found in the high-scoring groups, which insights teachers to consider the authority distributions within groups. Teachers can encourage students to more generate shared intellectual authority and contested authority in process of collaborative activities in classroom.

## **Appendix 4.1: Scores Rating Scale for CPS in Mathematics Outcomes in Middle School**

Scoring criteria: The high- and low-scoring groups in this study were assessed in absolute terms. The current absolute curriculum assessment in our schools is a percentage system, with a passing mark of 60 (60% of the total score). This system has been used in China's schools for roughly a century, for several reasons. First, it was developed in connection with an educational reform movement that took place in China in the late nineteenth and early twentieth centuries. Through this reform movement, Chinese education began to move away from the shackles of feudal education and towards modern education, as a result of learning from European and American education. Second, it was a result of convention. Therefore, as the maximum total score in this study was 10, a score greater than 6 identified the high-scoring group and less than 6 the low-scoring group.

## **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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**Chapter 5 The Development and Use of Opportunity to Learn (OTL) in the Collaborative Problem Solving: Evidence from Chinese Secondary Mathematics Classroom**

**Yinan Sun and Boran Yu**

## **5.1 Introduction**

In the 1960s, American educational psychologist Carroll proposed the concept of Opportunity to Learn (OTL) which focuses on whether students can gain sufficient practice experience in their learning (Carroll, 1963). Since then, great efforts were paid to define what counts as OTL and study OTL as the unit of learning. Prior studies mostly concerned the classroom level and believed OTL results from teachers' behaviour (Elliott & Bartlett, 2016). The assessment of OTL in PISA 2012 was also based on teachers' instructions and support (OECD, 2014). Analysing OTL benefits teachers by improving their teaching practices (Stevens, 1993). In addition to teachers' support, OTL could appear from interactions among students. Through collaborative learning, students can learn by explaining their thinking, collectively reflecting on the solution, and learning from peers, which could be seen as OTL provided for students (DeJarnette, 2018).

Raising students' ability to collaboratively solve problems is necessary in the context of the latest curricular reform, which implies the great value in studying how OTL develops and is used by students in this context. Firstly, the new curriculum emphasised that students should be the centre of classroom teaching while teachers provide guidance and support for their learning. Prior studies suggested the complex mechanism of students' learning in the classroom (Langer-Osuna et al., 2020) and

Y. Sun

101

Hefei No. 8 High School Education Group Shushan Branch, Hefei, China

B. Yu (B)

Beijing Normal University, Beijing, China e-mail<EMAIL_ADDRESS>
<sup>©</sup> The Author(s) 2024

Y. Cao (ed.), *Students' Collaborative Problem Solving in Mathematics Classrooms*, Perspectives on Rethinking and Reforming Education, https://doi.org/10.1007/978-981-99-7386-6\_5

thus it is of great importance to study their classroom learning performance. Understanding how OTL develops and is used by students would maximise students' selflearning. Secondly, one of the important aims of classroom learning is to gain the knowledge and develop the key competencies necessary to suit their further development in society. This aim has made studying how students gain and use OTL imperative for supporting students' sustainable development. Finally, collaborative problem solving provides students with an open environment where they discuss the problem with each other to develop a solution on which everyone agrees through different phases (Salminen-Saari et al., 2021). Compared with traditional teachercentred classroom teaching, collaborative problem-solving teaching allows students to express themselves more and thus would develop more OTL through student interactions.

This study investigated how OTL develops in the context of collaborative problem solving and how students use it. Firstly, we analysed the types of interactions students made and the discourse they used to figure out how OTL develops in collaborative problem-solving context. Two students failed to join in problem-solving tasks and their groups were analysed to figure out how they interacted with other group members how interactions could generate OTL and why they failed to seize the OTL developed from these interactions.

## **5.2 Literature Review**

Since the concept of OTL was first proposed in the 1960s, scholars have tried to define it in different ways. Allwright divided OTL into three categories: opportunities to input, opportunities to exercise, and opportunities to manage (Allwright et al., 1991). This categorisation was accused to be oversimplified and it has been proposed that OTL should be divided into the opportunities to input, opportunities to output, opportunities to interact, opportunities to feedback, opportunities to practice repeatedly, opportunities to understand discourse, and opportunities to understand OTL (Mao, 2016).

Past studies have paid great attention to how students take advantage of OTL, how teachers could provide more OTL for students, and the assessment of OTL. A large proportion of studies of OTL in the classroom focus on OTL in online learning, the inequalities in OTL brought by racial or spatial differences, and the relationship between teachers' attitudes towards teaching and students' OTL. Additionally, several studies focus on the macro-level of students' OTL (e.g., tuition fees, access to education) (Wang, 2018). While these studies have explored how external factors affect students' OTL, less is known about how students themselves create and use OTL.

Studies evaluating OTL aim to delineate how teachers provide students with important learning resources. Scholars used different ways to assess OTL, such as discourse analysis, questionnaire, interview, classroom observation, and field notes. Reeves and colleagues used text analysis and classroom surveys to analyse African mathematics classroom videos. They evaluated the extent to which students accomplished classroom tasks to compare OTL use in mathematics classrooms in different countries (Reeves et al., 2013). Wang followed the OTL evaluation framework proposed by Stevens and designed a questionnaire that surveyed the content coverage, content exposure, teaching emphasis, and instructional delivery quality. Based on questionnaire results, he further analysed the relationship between students' achievements and the OTL they received (Wang, 1998). Herman et al. (2000) developed an OTL evaluation framework consisting of four dimensions classroom content, teaching strategies, teaching resources, and assessment—based on teacher and student questionnaires and interviews. Other domestic scholars evaluated OTL from ten dimensions (including aim, preparation, meaning, etc.) and found that secondary students could feel the opportunities in method and those in difference but found it difficult to feel opportunities in meaning and those in challenges (Yin, 2018). Hao used a questionnaire to study whether OTL appeared in classroom teaching equally (Hao & Hu, 2016).

OTL has been studied as a key variable predicting learning results. Goodlad proposed that it is necessary to consider OTL as an important variable if one links learning results and the OTL gained by them (Goodlad et al., 1979). Freud thought that differences in academic achievement could be seen as differences in OTL. Studies suggest that students' academic achievement relates to OTL after controlling for students' learning ability and socioeconomic status (Vernon, 1971). Similar studies suggest that OTL predicts students' academic achievement more than teachers' skills, teaching efficiency, and expectations.

Few domestic studies have studied OTL in Chinese classroom settings. Most studies, using data from large-scale international surveys like PISA, conclude that OTL may lead to differences in students' learning achievement in different contexts. PISA surveyed students' OTL in three dimensions: (1) learning formal mathematics; (2) learning textual mathematics problems; and (3) learning applied mathematics. A re-analysis of PISA 2021 data suggested that Chinese students' mathematics performance is closely related to OTL, and that they had more opportunities to learn formal mathematics but fewer to learn applied and textual mathematics (Teng, 2014). Several researchers have analysed teacher-student interactions to study students' OTL (Bai & Lin, 2016).

Other studies have found a weak association between OTL and secondary students' academic performance but a stronger association between OTL and their non-academic achievements. They point out that self-learning plays an important role in learning, but students find it hard to get used to OTL provided by others. To sum up, the use of OTL is a possible learning result; thus, OTL is unequal to learning results. Additionally, while OTL use is closely related to students, it is also influenced by other factors like economics, resources, etc.

In the context of collaborative problem solving, OTL can influence students' learning. Students can respond to others in the collaborative learning process, but whether they can understand each other is uncertain. However, most studies have focused on how teachers' intervention can influence students' OTL. This gap implies the necessity of understanding how students interact to attain and use OTL.

## **5.3 Theoretical Framework and Method**

## *5.3.1 Commognitive Approach to Study Mathematics Learning*

Commognitive approach was originated from a learning science stance that conceptualize learning as participationist rather than acquisitionist (Sfard, 2007). This approach conceptualise learning as a form of communication and thus suggests that learning can be achieved by revising and expanding one's discourse. In the commognitive approach, learning mathematics means "to individualise the historically established discourse known as mathematical" (Chan & Sfard, 2020). Here, "Individualise" refers to being the agentive participant of this discourse. Specifically, students can not only follow its rules but also transform it flexibly according to their will and use it to inform the next step. Highly individualised discourse would become the primary media for one's thinking. In this case, learning is conceptualised as one's interactions with oneself. When it comes to case with more than one individual, learning is tantamount to conversations which have multiple channels and models not limited to verbal discussion. When two students are in a discussion, the discussion flow has three different channels (as shown in Fig. 5.1). There existed inter-personal channels where two individuals are interacting with each other. Intrapersonal channels, however, allows interactions with oneself to promote both individuals' learning. Among these channels, only the inter-personal channel is observable, while the intra-personal is hidden from observers. This commognitive approach has been widely used in empirical studies examining collaborative learning because of its emphasis on the dynamic interactions in learning. In this study, we believe that OTL is developed from these two forms of interactions and three different channels among group members in the four-student groups' discussions.

**Fig. 5.1** Multiple channels in interactions

**Table 5.1** Types of opportunity to learn (OTL)

## *5.3.2 Opportunity to Learn (OTL)*

Chan and Sfard (2020) generalised what kinds of interactions could count as OTL and in what circumstances OTL will develop. They suggested that there are two types of OTL, one of which is generated from a change in the initiative of students' discourse. This change can give learners more opportunities to participate in interactions with mathematical discourse. This interaction, where learners talking about mathematical objects, could be called mathematizing. OTL will develop when learners interact and talk about mathematical objects or subjectify these objects. Here, subjectifying means students articultating their dispositions about mathematics. The development of the other type of OTL needs a change in the type of discourse itself rather than in the initiative of discourse. These OTL may appear in object-level or meta-level discourse or when learners meet problems that need to be verified. Additionally, to use these OTL, students must actively participate in group discussions by expressing themselves or listening to others' explanations. These OTL could also be developed when students get in touch with language at different levels and have conflicts with others. The Table 5.1 generalises the two types of OTL and their characteristics.

## *5.3.3 Sampling and Data Collection*

We used videos of collaborative problem solving in a secondary mathematics class in Beijing from the Social Essential of Learning (SEL) project. In this class, the collaborative problem-solving task was Taks A, "Xiaoming's Apartment" (see Appendix for more details). A detailed investigation of interactions within the group is critical to understand how students can take advantage of OTL in collaborative learning activities. Rather than focusing on the successful learning cases, this study focused on students who failed to use OTL to reveal why students learn little in collaborative problem-solving. After carefully reviewing numerous collaborative problem-solving videos, two students and their groups were selected for further analysis to explore how OTL develops and benefits students through their interactions. Specifically, we found that some students got involved successfully initially but gradually got lost in the group discussion and further identified these groups for in-depth investigation. Each group included four students (two boys and two girls), coded separately as S1, S2, S3, and S4. Two characteristics were identified in these two groups:

## *5.3.4 Data Analysis*

Videos were transcribed for further analysis. Firstly, to explore the discourse used in collaborative problem solving, we divided students' discourse into two categories: mathematizing and subjectifying. There were two levels within the mathematical category: meta-level and object-level. The object-level concerns students' talking about mathematical objects while the meta-level concerns students' reflection on their talking about mathematical objects. Table 5.2 presents the coding scheme of students' discourse. We used different coloured circles with arrows to represent the type of discourse and direction of students' interactions. As shown in Fig. 5.2, black circles refer to students' discourse at the object level, meaning their discussions about mathematics itself. Grey circles refer to students' discourse at the meta level, meaning their thoughts or comments on some mathematical objects. White circles refer to students' discourse at the subject level, meaning their discussion about the learners themselves. Arrows show the types of interaction. Circles with arrows pointing to the bottom-left represent inter-person proactive interactions (students' initiatively discussing with others), while those with arrows pointing to the upperleft represent inter-personal reactive interactions (students' responding to others), and those with arrows pointing to the bottom represent intra-personal interactions (students' talking to themselves). After trial coding and a refinement of the coding scheme, two researchers independently coded the data, attaining an 86% consistency rate.

Additionally, we calculated the proportion of different types of discourse (Fig. 5.3) and the proportion of different types of interactions (Fig. 5.4). Considering there are four students within each group, we identified two different roles students are playing: leaders and followers. Leaders are charactersied with more mathematising discourse and more inter-personal interactions (e.g., S3 in the example). In this study, we focus on the interactions between focus students and leader students as such interactions could generate more OTL by getting focus students involved in collaborative learning. Although OTL appears in interactions indifferent forms and contents, we followed the definition by Chan and Sfard (2020) that captured OTL based on how discourse was developed and changed through students' interactions.

After coding the discourse and determining each student's role, we conducted a detailed analysis of the focus students' discourse to determine how they interacted

**Table 5.2** Coding scheme of students' discourse

**Fig. 5.3** The discourse types of students (example)

with other students and caught the OTL created from these interactions. The next section introduces each group's collaborative problem-solving process and discusses the development of OTL. Figure 5.5 outlines the data analysis process in this study.

**Fig. 5.4** The interaction types of students (example)

**Fig. 5.5** Data analysis diagram of this study

## **5.4 Results**

## *5.4.1 Group 1*

Group 1 had two boys and two girls. S2 was the focus student and participated less actively through the collaborative problem-solving process. At the beginning of the discussion, S2 interacted with others very well. As the discussion continued, however, S2's opinions began to differ from other group members', leading to conflicts. As a result, S2 missed several OTL generated from interactions among other group members, making it harder for him to follow later parts of the discussion.

#### **<IP_ADDRESS> The Stages of Discussion**

We divided the collaborative problem-solving process into three stages based on S2's performance and the group discussion content.

**Stage 1**: At this stage, group members mainly discussed the structure of the apartment and what one square metre looks like. However, S2 had vastly different opinions, and conflicts continually emerged. For example, when S4 proposed that the bathroom should be two square metres, S2 questioned her idea immediately, saying it was "impossible." When S4 amended her idea and said, "five square metres should be enough," S2 continued to reject it, declaring, "five square metres could never be enough." From what S2 said, it was apparent that he wanted to join the group discussion by rejecting others' views and gaining their attention. However, other students chose to ignore his rejections and went on with their original discussion. Finally, S2 stopped rejecting others (under the teacher's guidance), and the group entered the next stage of discussion.

**Stage 2**: At this stage, group members discussed the overall arrangement of the apartment, including what rooms the apartment had and how to furnish them. S2 did not keep pace with the others in. While the group discussed how to divide the apartment into different rooms, S2 considered how to draw the apartment or proposed irrelevant requests.  Whenever S2 tried to start a topic, the others ignored or rejected him, leading to his failure to get involved in the group discussion.

**Stage 3**: At this stage, group members finished most of their prior discussions and started drawing the apartment's arrangement on paper. S2 barely had any chance to participate in the discussion and instead just talked to himself. When other students decided to use a scale of 1:100, S2 said "it is irrational to use a scale of 1:1," showing he had not paid attention to what the others had said. Later, when the group discussion focused on how to draw the kitchen, bathroom, and living room, S2 did not express an opinion; however, when the other members had almost finished the drawing, S2 began to question the result, again saying something like "this is impossible in real life."

Overall, the focus student's performance followed some patterns at different stages of the group discussion. The group discussion went on in an organised way. The group firstly focused on what an apartment should have (e.g., a kitchen, bathroom, and living room) and how large one square metre was, which was necessary preparation work for drawing the apartment's floor plan. Then they moved to the overall arrangement of the apartment, the size of each room, and finally finished the task. At the beginning of the group discussion, the focus student kept questioning and explaining his ideas. But as the discussion continued, the focus student did not keep pace with others; instead, he thought about drawing (a task that should be done later) while his peers are discussing the arrangement. It is apparent that he wanted to take the initiative in the discussion but failed due to a mismatch between his ideas and those shared by the group. Finally, he lost interests and missed many of the ongoing discussion's details, leading to a situation where he could only speak to himself.

#### **<IP_ADDRESS> Analysis of OTL**

OTL developed in all three stages mentioned above through different types of discourse and interactions (see Table 5.3 for more details).

**OTL in Stage 1**: The first OTL appeared when the group was discussing the area of the bathroom. When S4 proposed that the bathroom could be two square metres, the focus student of this group (S2) questioned this idea, responding that it was "impossible." In this interaction, their type of discourse transformed from the object level to the meta level with S2's detailed explanation ("Because you need enough space to take a shower"). This explanation contributed to the development of the second OTL.

The third OTL was also developed while discussing the area of the bathroom. When S4 tried to adjust the bathroom to five square metres ("Actually, five square metres are enough"), S2 still rejected this new idea ("five square metres could never be enough"), which caused conflict between S2 and S4 again. In this conflicted interaction, the type of discourse transformed from the subject level to the meta level. The OTL did not work because S2 did not explain why he thought five square metres are not enough or suggest how many square metre should be enough.

The fourth OTL appeared from the discussion of the area of four bricks. When S3 suggested the area to be 36 square decimetres, S2 objected again, much as in the first OTL. The fifth OTL happened when the group was discussing the area of the bathroom under the teacher's guidance. When the teacher checked on how Group 1 was progressing, S3 mentioned that they could not confirm the area of the bathroom. S2 followed S3's question and asked, "How many square metres does a bathroom generally have?", which produced OTL through questions.

It is noticeable that the conflicts between S2 and the other students were partly caused by the difference in the type of their discourse. More specifically, S2's discourse was mostly at the meta level (e.g., "impossible"), while the students interacting with him turned the direction back to the object level. This could be seen as a latent conflict in the discourse structure.

**Table 5.3** A summary of OTL in Group 1

**OTL in Stage 2**: The sixth OTL appeared when the group discussed the apartment's overall structure. S2 thought it better to draw their ideas on the paper first, while S3 disagreed ("We can discuss first and then draw on the paper…. You do not have to draw now"). At this time, conflicts appeared, and the discourse turned from the object level to the subjectifying level.

The seventh OTL appeared when discussing whether a bathroom should have a washing machine, as S3 suggested. S2 rejected this idea and sought to regain the interaction initiative ("Could you please listen to me first?"), moving the content of their interaction from the object level to the subjectifying level.

The eighth OTL happened when the group discussed the kitchen area. When the group discussed whether the kitchen area could be eight square metres, S2 did not pay full attention but questioned whether it should be smaller, directing the discussion content from the object level to the meta level.

The ninth OTL appeared while discussing the area of the bedroom, which S2 thought should be less than ten square metres, while others disagreed.

At this stage, S2's discourse mainly aimed at returning to the group discussion by gaining the discourse initiative. However, his attempts did not always keep pace with the other members, leading them to ignore or disapprove of his ideas and causing his failure to participate in the group discussion. For example, when S2 said, "Could you please listen to me first?" S3 choose instead to discuss whether a bathroom should have a washing machine.

**OTL in Stage 3**: The tenth OTL appeared when choosing the scale. When S4 decided to use the scale of 1:100 ("We can use 1 cm to represent 1 m."), S2 misheard the scale to be 1:1 and rejected S4's idea again ("1:1 is impossible."). It is apparent that S2 did not pay full attention to the others, leading to conflicts and turning the discourse type from the object level to the meta level.

The eleventh OTL emerged at the end of the discussion. When the group finished drawing, the focus student almost gave up on joining the discussion. S2 still made several criticisms ("This design is impossible"), but his ideas were again ignored.

Generally, the group's discussion followed an organised order, proceeding from discussing the apartment's overall structure to allocating the rooms and drawing the floor plan. However, the focus student consistently failed to follow the pace taken by the group. For example, the focus student wanted to draw when others discussed each room's area. Additionally, the focus student's ideas were always rejected or ignored, making him miss OTL.

## *5.4.2 Group 2*

#### **<IP_ADDRESS> The Stages of Discussion**

Based on the performance of S2 and the group discussion content, we divide the collaborative problem-solving process into three stages.

**Stage 1**: At this stage, the group focused on the general arrangement of the apartment and details about the kitchen and bedroom. In this process, S2 took a very active part in group discussion and tried to express his own ideas (e.g., "I think the living room should be here, and there should be three bedrooms next to the living room," "I know what questions you have"). However, S2 did not get an active response from other students. For example, when S2 proposed the idea of three bedrooms, S3 rejected it immediately ("It is insane to have three bedrooms in 60-square-metres").

The group spent a relatively long time discussing the area of the living room and the bedroom. When S2 proposed that he knew the key to solving the question, other students refused to give him the discourse initiative ("Could you stop talking?"). In the later discussion, S2 tried to express himself and actively join the discussion, but the other students seldom actively responded to him.

**Stage 2**: At this stage, the group had a general discussion on the area of each room. After repeatedly receiving negative feedback from other members, the focus student gradually lost enthusiasm for joining the group discussion. When the group had a heated debate about the area of specific rooms, S2 just echoed what others said and did not express his own ideas.

**Stage 3**: The main task in this stage was diagramming the apartment they had designed. S2 tried to rejoin the group discussion by initiating more interactions. When the group was discussing how to draw the graph, S2 proposed his ideas ("I think we can…," "Listen to me, there is a simpler way that…"). However, the other students ignored him and discussed whether the bathroom and the toilet should be separate. When S2 asked, "Why do you split the bathroom and the toilet? They could be put together," other members did not explain their thoughts and passed over this question ("He splits the two rooms, not me.").

Although the focus student joined the group discussion more actively at this stage, he did not contribute any constructive ideas. In effect, the other students formed a "smaller sub-team" to continue their discussion, leading to S2's missing many OTL that emerged from their interactions.

The focus students in the two groups had similar characteristics in the extent to which they actively participated in their group's discussion. At the beginning of the group discussion, the focus student of Group 2 did not actively participate. It was apparent that he wanted to join the group learning at the second stage of discussion, but failed because he had paid little attention to other students' ideas, leading them to ignore his. Even in the last stage, other students paid no attention to what the focus student said, which led to his missing many OTL.

#### **<IP_ADDRESS> Analysis of OTL**

OTL developed in all of the above-mentioned three stages through different types of discourse and interactions (see Table 5.4 for more details).

**OTL in Stage 1**: OTL appeared four times in the first stage. The first OTL appeared during the discussion of the position of the living room and the bedroom. When the focus student proposed his idea ("I think the living room should be here and there are three bedrooms around the living room, you can put it"), other members rejected it, believing it was impossible to accommodate three bedrooms in a 60-square-metre apartment. Although he later tried to explain why there should be three bedrooms, the other students did not give him the chance, which made him lose this OTL. In this OTL, the type of discourse stayed at the object level.

The second OTL appeared in the discussion of the area of the apartment. The focus student suggested that he had found the key to solving the problem, but the others chose to ignore him. This negative response made it difficult for S2 to seize OTL. The type of discourse in this OTL moved from the object level to the subjectifying level.

The third OTL appeared when they were discussing the number of bedrooms. When S3 stated it was impossible to have three bedrooms in the apartment, the focus student questioned this ("Having three bedrooms could be possible."); the other students did not follow their discussion. At this time, the type of discourse changed from the object level to the meta level.

**Table 5.4** A summary of OTL in Group 2

**Table 5.5** A comparison of the discourse type of the follower and leader in Group 1

The fourth OTL was in their discussion of how many floors an apartment should have. While the other students thought an apartment should have one or two floors, S2 thought one should have three or four. However, S2's questioning was still ignored by the other members ("Could you stop talking for now?"), which led to his failure to seize this OTL.

**Table 5.6** A comparison of the discourse type of the follower and leader in Group 2

**OTL in Stage 2**: The next three OTL emerged in the second stage. The fifth OTL appeared when the group discussed "whether a kitchen should have a door?" The focus student believed a kitchen does not need a door ("Why does a kitchen have to have a door?"), but the other group members said the "door" was simply the room's exit (on the picture), not a real one. The type of discourse stayed at the object level in this discussion.

The sixth OTL appeared when the group discussed each room's area to calculate the apartment's overall area. S4 thought the bathroom was similar to a half-circle and thus proposed calculating the area of a half-circle first. However, the focus student diverged from others and urged calculating the overall area based on the apartment structure rather than the area of each room. He refuted S4's idea ("We could calculate the overall area by looking at the one enclosed by the edge. Why do we have to calculate each one?"). This mismatch led to the other students' ignoring the focus student's idea, following their shared one, and continuing to discuss the area of the bathroom. The type of discourse turned from the object level to the meta level.

The seventh OTL appeared when the focus student proposed an idea as the group discussed the area of the main bedroom ("The main bedroom should be at most half of the one you drew."). This led the other group members to end this part of the discussion and start discussing the other four rooms. Their discourse type stayed at the object level.

**OTL in Stage 3**: The eighth OTL appeared while the group worked on the apartment floor plan. When the focus student tried to share his thoughts to initiate the discussion ("I had a simpler way to draw it that…"), the other members paid no attention and focused on discussing whether the bathroom and toilet should be placed together. The discourse type in this discussion turned from the object level to the subjectifying level.

The ninth OTL appeared when the group calculated the area of the bathroom. The focus student proposed his own calculation method ("We can directly calculate the area of this") but was rejected by the other members ("It will not work"). The discourse type of this short discussion changed from the object level to the subjectifying level. Due to repeated negative feedback, the focus student gradually stopped sharing his ideas.

The focus student in this group had some specific characteristics. In the first stage, the focus student did not participate actively but still tried to join the others. However, the focus student failed to integrate into the group due to negative feedback, leading to an inactive participation. Although the focus student tried to rejoin the group in the third stage, he did not succeed until the end of the discussion.

## **5.5 Discussion**

## *5.5.1 Type of Discourse*

In the collaborative problem-solving process, the proportion of mathematizing discourse used by the focus student in Group 1 first increased then decreased later, while the proportion of his subjectifying language remained at a relatively high level. For the focus student of Group 2, the proportion of mathematizing discourse was low at first but increased later, while the proportion of his subjectifying discourse increased at first but decreased later.

The focus student of Group 1 actively participated in the group discussion in the first stage and constantly tried to initiate discussion in the second stage to get back to the group discussion. However, the other members rejected or ignored his ideas as he is not at the same page, leading to his decreased participation in the third stage. Therefore, while the focus student's involvement was first active, this activeness decreased later, making him miss many OTL. We also found that the focus student in group 1 used mathematizing discourse more at the beginning of each discussion stage and subjectifying discourse more at the end of each satge.

The focus student of Group 2 was similar to the one of Group 1. Both actively participated in the group discussion in the some stages and tried to initiate the discussion, while the other students rejected their ideas. The difference lies in that Group 2 focus student did not actively join the discussion at the first beginning but tried to rejoin in later stages. The negative feedback from others in Group 2 made focus student failed to get involved in the second stage. In the third stage, the focus student tried to rejoin the group discussion and became more active. His inactive involvement in the first stage and failure to re-involve in later stages in the group discussion made him miss more OTL. Like his counterpart in Group 1, Group 2's focus student used mathematizing discourse more at the beginning of each discussion stage and subjectifying discourse more at the end.

A detailed analysis of the discourse type suggests that the proportion of different types of discourses may influence students' involvement in collaborative problem solving and further their use of OTL. More specifically, the more mathematizing discourse and less subjectifying discourse students used, the more active they were in the group discussion and the more OTL they could gain. On the contrary, if students use more subjectifying discourse and less mathematizing discourse, they might find it hard to join the group discussion and miss several OTL.

Table 5.6 and Table 5.7 compare the proportion of different types of discourse used by the leader student and focus student (who played the follower role) in each group. For Group 1, S2 was the focus student and follower, while S3 led the collaborative problem-solving process, using the largest proportion of mathematizing discourse in the group. As shown in Table 5.6, the follower student and leader student influenced the proportion of the two types of discourse each other used: when the leader student used less mathematizing discourse, the follower student may use more. The same pattern appeared in their use of subjectifying discourse. This pattern may suggest that a mismatch in the discourse types could contribute to a failure in getting involved in collaborative problem-solving.

For Group 2, S2 was the focus student and follower, while S4 was the leader of the collaborative problem-solving process. Table 5.6 shows that in the first two stages, the discourse used by the leader student and follower student also influenced each other. As the third stage lasted a rather long time, both used a large portion of mathematizing and subjectifying discourse; however, the leader student still used less subjectifying discourse. Since the goal of group collaboration is to solve a mathematics problem, a extremely large proportion of subjectifying discourse could distract the flow of group discussion.

Figure 5.6 shows the relationship between the students' different roles in collaborative problem-solving and their use of discourse in different types. The research results suggest the following two conclusions:

The leader student and follower student could influence the proportion of different types of discourse that each other used. When the leader student used more mathematizing (or subjectifying) discourse, the follower student may use less.

Type of discourse

**Fig. 5.6** The relationship between students' roles in the group and their discourse types

Several reasons for students' failure in participation were recognised from this perspective. Firstly, rather than trying to use more mathematizing discourse, the focus student used more subjectifying discourse to express himself, making it harder for him to communicate with others. Secondly, the focus student did not pay sufficient attention to the discourse used by the leader student or other students but tried to distract them, which prevented him from seizing and using OTL.

## *5.5.2 Type of Interactions*

The type of interactions also changed in different stages (for details, see Group 1 in Table 5.7 and Group 2 in Table 5.8). For the focus student in Group 1, the proportion of intra-personal interactions decreased first and increased later, while the proportion of inter-personal interactions increased first and decreased later. In the first stage, the focus student interacted with other group members less and with himself more. In the second and third stages, these intra-personal interactions became less frequent and inter-personal interactions more frequent.

For the focus student in Group 2, the proportion of intra-personal interactions increased first and decreased later, while the proportion of inter-personal decreased first and increased later. In the first and the last stages, the proportions of these two types of interaction were essentially the same, while the focus student interacted with himself more and with others less in the second stage.

These changes suggest that the type of interaction may influence how students become involved in collaborative problem solving and whether they can use OTL. More specifically, the more students interacted with others, the more active they were in the group discussion, and the more OTL they took advantage of. In contrast,

**Table 5.7** A comparison of the interaction type of the follower and leader in Group 1

**Table 5.8** A comparison of the interaction type of the follower and leader in Group 2

the more students interacted with themselves, the less active they were in the group discussion and the more OTL they missed.

Tables 5.7 and 5.8 also compare the types of interactions by the leader student and focus student (a follower student) in Group 1 and Group 2. For Group 1, the proportion of proactive interactions S3 (the leader student) made continually increased, while S2 (the focus student and follower student) made more such interactions and less later. The frequency of S2's and S3's intra-personal interactions first increased and then decreased. Generally, the leader student made more inter-personal interactions (whether proactive or inactive), while the focus student made more intra-personal interactions. This could suggest that the leader student tended to interact with others, while the focus student (as the follower) tended to interact with himself.

For Group 2, the proportion of proactive interactions that S4 (the leader student) and S2 (the focus student and the follower student) made decreased first and increased later, while their proportions of inactive ones kept decreasing. S4 made more and more intra-personal interactions, while the frequency of these interactions S2 made increased first and decreased later. Similar to Group 1, the leader student interacted more with others while the focus student interacted more with himself.

Figure 5.7 shows the relationship between different roles in the group problem solving and the interaction type made by these students. The research results suggest the following two conclusions:

(1) The interaction type can influence the extent to which students participate in the group discussion and further their success in seizing OTL. Although intrapersonal interactions can help students think independently, more inter-personal interactions can help involve them in collaborative problem solving.

**Fig. 5.7** The relationship between students' role in the group and the interaction type

(2) Students in different roles had different interaction type preferences. Leader students preferred interacting with others, while follower students preferred interacting with themselves.

Several reasons for students' failure in participation were recognised from this perspective. Firstly, the focus students made less interactions with others but always communicated with themselves. This excluded them from the group discussions and prevented them from integrating themselves into the group discussion. Secondly, the focus students did not follow the type of interactions made by the leader student as well as other members.When the leader students discussed ideas with other members, the focus students interacted with themselves. When the leader students and other members thought independently and interacted with themselves, the focus students tried to interact with others. However, by this time, the inter-personal interactions had nearly ended, leading to the focus students' keeping missing OTL.

## *5.5.3 OTL and the Role of Student*

We identified the leader student, follower student (focus student), and other members in each collaborative problem-solving group. It is suggested that OTL development is closely related to leader students. In the collaborative work process, the follower students conflicted with the leader students by explaining, questioning, and taking the discussion initiative. While these conflicts can create OTL, the focus students missed the emerging OTL by being ignored and failed to participate in the group discussions. Figure 5.8 shows a more systematic representation of how students in different roles interacted to develop OTL. The analysis of discourse types and interaction types in the previous two sections suggests that the transformation between different discourse types and the new conversation contexts generated by the leader students could be

**Fig. 5.8** The process of OTL development

the foundation for developing OTL. The follower students' explanations, questions, initiations, and conflicts further developed OTL. Several mismatches between focus students and leader students were summarised below from the perspective of OTL:

## *5.5.4 A Summary of Factors that Influence the Development of OTL*

This chapter presents a detailed discussion of the different discourse and interaction types among students in different group roles. Qualitative analysis suggests that students' discourse and interactions are related to OTL in a complex way, not a simple linear manner. More specifically, using mathematizing discourse and making inter-personal interactions promoted the extent to which students participated in the group discussion.

The discourse and interactions were also closely related to the leader students, focus students (usually a follower student who failed to engage in collaborative work), and other group members. This complex interactive relationship suggests that many factors influence OTL development. In the collaborative work on solving mathematics problem, discourse and interaction types differentiate students' roles in group discussions. Students who used more mathematizing discourse and made more proactive interactions were identified the leader students. We assume that these leader students are talking more about mathematics and communicating with others to promote the problem-solving. A change in the leader students' discourse type could facilitate OTL development. However, the alignment between leader students and focus students decided whether this OTL could be created successfully and further influenced whether the focus students could benefit from it.

In other words, the leader students provided OTL by changing the discourse type or initiating new discourse. Later, the followers could question or challenge leader students to create conflicts, which could further develop OTL. The followers could also develop OTL alone by explaining their ideas to others or initiating a discussion. Therefore, overuse of subjectifying discourse and frequently interacting with themselves were two main reasons the focus students (follower students) failed to participate in the group discussion. Figure 5.9 shows how different factors can influence OTL in collaborative problem solving.

## **5.6 Conclusion and Implications**

This study has explored the development of OTL in collaborative problem solving in a secondary mathematics class. Instead of paying a close attention to successful cases, we focused on two groups where there is a focus students who failed to engage in group work and further to seize the OTL. By analysing students' discourse and interactions, we identified different roles students played and closely examined the interactions between leader students and focus students to study how OTL can be developed, seized, and used. We further conceptualised how these factors could influence the development and use of OTL as a supplement to the current theoretical conceptualisation of the mechanism of collaborative learning.

**Fig. 5.9** A systematic review of how different factors influence the development and use of OTL

Results suggest that students' discourse and interactions are closely related to OTL development, as they can influence the extent to which students actively participate in group discussions and further their learning results. This is aligned with the nature of collaborative learning as social interactions that are highlighted by past studies (Palincsar & Herrenkohl, 2002). Specially, we found that students with less mathematizing discourse, more subjectifying discourse, and more intra-personal interactions are hard to participate in collaboration and thus fail to take advantage of OTL. According to Smith and MacGregor (1992), peer discussion and peer tutoring are the most common forms of collaborative learning, which requires an in-depth involvement in interactions. Despite the social nature of collaborative learning, Laal and Ghodsi (2012) also emphasised the academic nature of collaborative learning. This aligns with our findings that students used less mathematizing discourse would fail in get involved in group discussion to learn from their peers.

This study has several practical implications for teachers in the teaching of collaborative problem solving. Collaborative problem-solving teaching provides an open atmosphere for students' learning, one that allows them to communicate with each other freely and elaborate on their advantages. Recent studies have revealed multiple factors that may influence such socialization culture (Dong & Kang, 2022), for example. Students are the agents of this sociocultural learning, while teachers are facilitators who provide guidance for students' learning. However, as teachers do not assign each student a discussion role but let them take on different roles naturally, leader students may not realise their responsibility and thus may make other group members miss many OTL. Therefore, teachers could assign different group discussion roles to students so that they can discuss ideas in a more organised way and effectively use more OTL.

## **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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## **Chapter 6 The Characteristics of Mathematical Communication in Secondary School Students' Collaborative Problem Solving**

**Jie Yang**

## **6.1 Introduction**

Researchers and teachers have gradually realised mathematical communication is an indispensable part of mathematics classroom teaching and learning. Through mathematical communication, teachers share mathematical knowledge and methods, while students improve their understanding of mathematical concepts. In 1989, mathematical communication ability was mentioned in *Curriculum and Evaluation Standards for School Mathematics* (National Council of Teachers of Mathematics, 1989); since then, different countries have added mathematical communication to their curriculum standards and syllabi (Sun, 2003). Communication has been a key ability in the PISA 2000–2012 assessment frameworks, with the PISA2021 assessment framework identifying it as an important twenty-first-century skill (Cao & Zhu, 2019). Mathematical communication has become the focus of mathematics classes (Neria & Amit, 2004).

Chinese researchers and teachers have focused on mathematical communication in classes for rather a long time. Chen (1990) introduced American mathematics curriculum standards, which included the requirement that students "learn to communicate in mathematical language." Shen et al. (1991) published relevant literature on the development of mathematical communication ability.

Since the beginning of the twenty-first century, China has gradually added mathematical expression and mathematics communication requirements to mathematics curriculum standards for compulsory education and senior middle school. Both the *Mathematics Curriculum Standard for Compulsory Education (2011)* (Ministry of Education of the People's Republic of China, 2012) and *Mathematics Curriculum Standard for Compulsory Education (2017)* (Ministry of Education of the People's Republic of China, 2018) repeatedly emphasised the importance of mathematical

J. Yang (B)

Zhenchuan Senior High School, Suzhou, Jiangsu, China e-mail<EMAIL_ADDRESS>
<sup>©</sup> The Author(s) 2024

Y. Cao (ed.), *Students' Collaborative Problem Solving in Mathematics Classrooms*, Perspectives on Rethinking and Reforming Education, https://doi.org/10.1007/978-981-99-7386-6\_6

expression and mathematical communication and highlighted the significant value of increasing mathematical communication in classroom teaching. Although research on mathematical communication in social and cognitive aspects has been conducted (e.g. Langer-Osuna et al., 2020), mathematical communication has become a key issue in Chinese mathematics education contexts.

Open-ended tasks could provide affordances for communication (Chan & Clarke, 2017). They can stimulate students to engage in activities in classroom and encourage students to explore and investigate (Osana et al., 2006). Teacher using open-ended tasks and asking students to explain their process and reasoning are beneficial to mathematical communication (Cai et al., 1996). Mathematical communication has attracted considerable attention from scholars worldwide, and studies suggest that Chinese teachers and students have started to pay attention to mathematical communication in their classes (Xu & Wang, 2017). However, various factors may interfere with the quality of students' mathematical communications in mathematics classroom. For example, students with poor communication skills would find it difficult to communicate with each other and express their opinions clearly, might not be patient enough to communicate, etc. (Su, 2003). Chinese students' mathematical communication ability still needs to be improved (Shen, 2014). Since the current situation, systematic research on students' mathematical communication in case of collaboration on open-ended tasks is necessary.

This study focused on the characteristics of mathematical communication in secondary school students' collaborative problem solving to address the following research questions:

## **6.2 Research Methods**

## *6.2.1 Data Selection*

The data for this study were collected from eight Grade 7 classes from two junior high schools, LH and YC, in Beijing by the research team during the 2017–2018 school year. Each class had six to eight valid groups, with videos and task sheets being collected from 56 groups in total.

The collaborative problem-solving task used in the study is shown in Appendix Task (1) To quantitatively evaluate the results of students' collaborative problem solving, the task scoring criteria were formulated as shown in Table 6.1.

According to the scoring criteria determined in Table 6.1, the results of 56 teams' collaborative problem solving were scored. The scoring results are shown in Fig. 6.1.

**Table 6.1** Scoring criteria

**Fig. 6.1** Scoring results

Eight groups were selected as research subjects in this study to analyse students' mathematical communication in a focused way, as shown in Table 6.2. Specifically, this study plans to select a group with a score higher than 3.5 and a group with a score lower than 2.5 in each teacher's class. Groups with a score higher than 3.5 were named high-scoring groups, and those with a score lower than 2.5 points low-scoring groups. As all the groups in Teacher A's classes scored 2.5 points or more, two highscoring and two low-scoring groups were selected from the two classes taught by Teacher B (in the same school as Teacher A).

This selection method minimised the influence of schools and teachers on the coding results. In similar external environments, group scores are significantly different. Therefore, the differences in high-scoring and low-scoring groups' mathematical communication characteristics could be better compared by analysing these groups.

**Table 6.2** Participations

## *6.2.2 Data Analysis*

After data collection, this study constructed the sets of elements in students' mathematical communication process based on existing literature. The following formula.

$$\begin{aligned} A\_1 &= \{E\_1, E\_2, \dots, E\_{n1}\}, & A\_2 &= \{E\_{n1+1}, E\_{n1+2}, \dots, E\_{n2}\}, \\ A\_3 &= \{E\_{n2+1}, E\_{n2+2}, \dots, E\_{n3}\} \end{aligned}$$

yielded the union of the above sets.

$$A = \{E\_1, E\_2, \dots, E\_n\}.$$

The videos of two groups were then selected for precoding, based on which we refined the element set mentioned above and deleted low-frequency elements from the coding results to construct an initial set of elements for mathematical communication in collaborative problem solving:

$$A' = \{E\_1, E\_2, \dots, E\_m\}.$$

To solve the problem of communication elements not belonging to the same level or overlapping each other, we established a more concise and efficient multi-level coding framework by seeking the patterns in students' mathematical communication and combining some elements into the same module.

Based on the established coding framework, this study coded students' mathematical communication processes in open-ended collaborative problem solving. Then, the researchers analysing the coded data and further summarised and constructed the main characteristics and process of secondary school students' mathematical communication in collaborative problem solving.

Additionally, we compared the differences between the characteristics of highscoring groups' and low-scoring groups' mathematical communication from two aspects. The first one is the number of communication elements. The second one is students' roles in mathematical communication, identifying each group's leaders, supervisors, marginalised students, and blockers.

## **6.3 Study Results**

## *6.3.1 Elements of Mathematical Communication in Secondary School Students' Collaborative Problems Solving*

The set of elements of mathematical communication was constructed based on the relevant literature. This study reviewed recent literature on collaborative problem solving and mathematical communication and summarised communication elements from them, as shown in Table 6.3.

As shown in Table 6.3, some communication elements did not belong to the same level, and there were obvious overlaps and repetitions. Therefore, these elements were re-sorted by disassembling some elements and merging similar elements for a more effective coding system. The communication elements were divided into two main parts, the cognitive dimension and the collaborative dimension, as shown in Table 6.4.

A total of 54 communication elements were sorted. However, not all belonged to the same level or overlapped each other, and so did not fully cover all the dialogues in the students' mathematical communication process. Thus, students' mathematical communication could not be effectively coded by these elements.

To solve the above issues, two videos were selected from the collected data to further revise the elements. The two videos were of Teacher A's Class 1 Group 1 from LH middle school and Teacher C's Class 5 Group 1 from YC middle school. In precoding these two groups, we tried to code every conversation between the students based on the existing elements. When it was found that some students' conversations could not be classified using any of the above 54 elements, new elements were added to the set. The precoding results are shown in Table 6.5.

Two new communication elements were added, namely "Refute" and "Propose a plan." Many elements did not appear in the precoding process. These elements were deleted to make the element set more suitable for the selected subjects. The improved element set contained 19 communication elements, as shown in Table 6.6.

The mathematical communication element set displayed in Table 6.6 was taken as the first-level coding framework for mathematical communication in this study.

The students' dialogue in the mathematical communication process had certain patterns and thus could be divided into certain communication modules. Sometimes these modules could be clearly divided, and sometimes there was no obvious boundary. Dialogues between students in different modules could occur simultaneously. For example, while S1 and S3 may have a dialogue on one topic and S2 and S4 may have a dialogue on another, both dialogues can still be roughly divided based on the students' communication topics. Thus, the first-level coding framework given in Table 6.6 can be merged into ten communication modules. The explanations for these ten communication modules and their patterns based on the first-level coding are shown in Table 6.7.

**Table 6.3** Elements of mathematical communication

**Table 6.4** Elements of mathematical communication (improved)

*Note* The "Artifact" refers to the specification process of students' thinking, including their experimenting by hand or writing answers down, while the meaning of this concept differs in different tasks

There were some similarities between the "negotiate," "argue" and "quarrel" modules, but also some obvious differences. Students would discuss an issue in the "negotiate" module, reach a consensus and promote the problem-solving process. For example:

Student 1: You don't have to draw anything special, just mark each room. What's it for? Five rooms, three bedrooms? Two bedrooms, two bedrooms, one living room, one kitchen and one bathroom.

Student 3: Two bedrooms and one living room, one kitchen and one bathroom.

Student 1: Just right, two bedrooms, one living room, one kitchen and one bathroom, just right.

Student 3: Two bedrooms, one living room, one kitchen and one bathroom, roughly the same.

In the above dialogue, Student 1 proposed that the five rooms should include two bedrooms, one living room, one kitchen and one bathroom. Student 3 responded to Student 1 and the two reached an agreement, promoting the problem-solving process.

**Table 6.5** Precoding results

*Note* "/" refers to the conversations that cannot be classified, including teachers' instructions or students' talking about irrelevant topics

**Table 6.6** Elements of mathematical communication (Improved)

In the "argue" module, students criticise, refute and question each other on specific issues before finally reaching a relatively unified consensus, which promotes the problem-solving process. For example:

Student 1: Just draw it like this. Draw a big square.

**Table 6.7** Modules of mathematical communication

Student 3: A square. Student 3: The room can't be square. Student 1: Right, rectangle, rectangle.

In the above dialogue, Student 1 proposed drawing a large square to represent the apartment, but Student 3 refuted Student 1's view, pointing out that the room could not be square. Student 1 agreed and they reached a relatively unified consensus, which promoted the problem-solving process.

Students also express their views, pose questions, refute and respond in "quarrel" modules. However, unlike in the "negotiate" and "argue" modules, they fail to reach an agreement. For example:

Student 4: Two bedrooms.

Student 2: But he lives alone.

Student 3: Why can't there be two bedrooms in one's apartment?

Student 2: Why don't you set a grocery store? It could be small.

Student 4: Xiao Ming's apartment.

Student 2: Then why don't you have a utility room? Isn't there usually a utility room at home?

Student 1: No one has a utility room. The utility room is combined with the toilet.

Student 4: Let's first determine what the five rooms are.

In the above dialogue, Student 4 proposed having two bedrooms, while Students 2 and 3 refuted Student 4. When no consensus was reached, Student 2 shifted the topic and suggested adding a utility room, at which point Student 1 refuted Student 2. However, the group then changed the topic, leaving the problem unresolved.

The ten mathematical communication modules displayed in Table 6.8 were taken as the second-level coding framework for mathematical communication, based on which the two groups of students' mathematical communication were coded. The results of the coding are as follows.

By integrating the analysis of the ten communication modules with Polya's fourstage problem-solving process theory, these modules can be further divided into four main parts: "Analyse the problem," "Make a plan," "Carry out the plan" and "Evaluate." The "Carry out a plan" part includes seven communication modules: "Provide information," "Negotiate," "Argue," "Quarrel," "Off-task discussion," "Calculate" and "Artifact."

Based on the postcoding process, a systematic and hierarchical three-level coding framework was finally formed, as shown in Table 6.9.

#### (I) **The characteristics of mathematical communication in secondary school students' collaborative problem solving**

The dialogues of the eight groups were coded based on the three-level coding framework given in Table 6.9. The first-level coding results are shown in Table 6.10. The numbers in Table 6.10 represent the frequency with which each element appeared in each group's mathematical communication.

**Table 6.8** Postcoding results

**Table 6.9** Three-level coding framework

As shown in Table 6.10, the "express views" element was the most frequent (300 times), indicating that students were willing to state their ideas in the communication process. It was followed by "pose a question," "question," "respond," "answer" and "artifact" elements, all appearing more than 150 times. The elements "provide information," "refute," "urge," "evaluate" and "approve" appeared more than 50 times, while "improve views," "explain," "correct errors," "pose a rhetorical question," "assign tasks," "propose a plan," "generalise" and "summarise" appeared fewer than 50 times.

The second-level coding results for the eight groups are shown in Table 6.11. The numbers in the table represent the frequency with which each module appeared in each group's mathematical communication process.

As shown in Table 6.11, the "quarrel" module was the most frequent (46 times), indicating it was often difficult for students to reach an agreement in the discussion process. The elements "argue," "artifact" and "negotiate" appeared 43, 22 and 20 times, respectively, while "off-task discussion" made ten appearances. "Analysing the problem," "make a plan," "provide information," "calculate" and "evaluate" each appeared less than ten times.

Thus, the main characteristics of mathematical communication in secondary school students' collaborative problem solving can be summarised as follows:

**Table 6.10** First-level coding results

*Note* HS means high-scoring and LS means low-scoring

#### (1) **"Negotiate," "argue," "quarrel" and "artifact" were the basic communication modules in mathematical communication**

The coding framework for this study contained ten communication modules. However, this study found that not every communication module appeared in each group's mathematical communication process. Only four modules—"negotiate," "argue," "quarrel" and "artifact"—appeared in all groups' discussions; the other six communication modules did not. Moreover, "negotiate," "argue," "quarrel" and

**Table 6.11** Second-level coding results

"artifact" were the most frequent modules in the mathematical communication process.

"Negotiate" and "argue" were the most common, with each group experiencing five to 15 rounds of "negotiate" and "argue" modules. "Negotiate" and "argue" were the two most important communication modules for promoting the problem-solving process.

"Argue" and "quarrel" modules both contained "express views," "refute" and "question" elements. However, unlike in the "argue" module, students in the "quarrel" module either could not reach a consistent conclusion or changed the topic before doing so, despite still discussing the problem to some extent. Although the students could not reach an agreement, the "quarrel" module promoted the problem-solving process to an extent and was undeniably an important basic communication module in the problem-solving process.

"Artifact" was an important module, enabling students to present discussion results on a task sheet and visualise the problem-solving results. Therefore, "artifact" was also a basic communication module in the problem-solving process.

#### (2) **Students were willing to express their views and criticise each other in the mathematical communication process**

Based on the data, "express views" and "pose a question" were the most frequent communication elements, appearing 300 and 219 times, respectively. This shows that students actively expressed themselves and raised some questions in the mathematical communication process. There was no situation where they had no idea what to do.

In addition, the frequencies of the "question" (180) and "refute" (91) elements were also considerable, indicating that students dared to question and refute others' opinions in the mathematical communication process.

One possible reason for this is that the students in each group were all from the same class and familiar with each other. Therefore, silence was rare, and the students were willing to engage in communication.

#### (3) **The students' mathematical communication process was always incomplete**

The data revealed that many groups lacked "analyse the problem," "make a plan," and "evaluate" modules in their mathematical communication process, with "analyse the problem" being the most conspicuous absence. Only two of the eight groups analysed the problem, while two groups did not make plans or evaluate.

Among the eight groups, only two experienced all four parts of mathematical communication, indicating the students were unfamiliar with the collaborative problem-solving process. As Chinese teachers pay little attention to the collaborative problem-solving process in traditional mathematics classes, students rarely have opportunities to use it and are often unclear about the steps involved. When the students in this study faced this unfamiliar teaching method, their teachers did not guide them to experience the necessary problem-solving parts.

#### (4) **There were many cases in which students failed to reach an agreement in the mathematical communication process**

The data revealed the "quarrel" module appeared most frequently (46 times) in the eight groups, while the "argue" (43) and "negotiate" (20) modules were relatively less frequent.

This suggests that the students dared to refute and criticise other students' views, but had difficulty reaching a consensus. Students often changed the topic before reaching an agreement and left many issues unsolved in the problem-solving process. These issues usually need to be discussed later, thus wasting a lot of time.

#### (5) **Discussions unrelated to the problem were common in the students' mathematical communication process**

The data revealed that off-task discussions inevitably occurred in the students' mathematics communication. In the eight groups, students had a total of ten unrelated discussions, indicating they were easily distracted during the problem-solving process. There were too many distractions in their complex classroom environment; students were easily disturbed by other groups, the recording equipment or the situation of the problem. For example, in Low-scoring Group 3, students had an off-task discussion:

Student 2: Your family is in the real estate business. Student 1: No. Student 2: Look over there. There's a camera. Student 3: Zhong, are you in the real estate business now? I mean now. Student 3: Actually, I really want to get into the real estate business.

In this dialogue, the students were influenced by the recording device and the problem situation, leading to off-task discussions.

#### (6) **There were fewer conversations in the collaborative dimension than in the cognitive dimension**

Data analysis revealed that students' dialogues involved the collaborative elements "urge," "evaluate," "approve" and "assign tasks" 60, 68, 123 and 45 times, respectively, for an average of 74. Several student dialogues involved all four collaboration elements, indicating they were concerned about group collaboration. However, on average, student dialogues involved each cognitive element more than 100 times, indicating that they paid more attention to promoting the problem-solving process cognitively than through group collaboration.

However, students also had a certain amount of dialogues on the four elements of the collaboration dimension, which also showed students' concern for group collaboration.

By combining the three-level coding framework for mathematical communication with the main characteristics of mathematical communication, this study established a model of students' mathematical communication process, as shown in Fig. 6.2.

#### (II) **The differences of mathematical communication characteristics between high-scoring and low-scoring groups**

As shown in Fig. 6.3, there were similarities between high-scoring and low-scoring groups in first-level elements, but also great differences in certain elements.

The number of dialogues involving the "respond" element was significantly higher in the high-scoring groups than in the low-scoring groups. This indicates that students with high scores were more active in responding to other students' opinions and expressing their own.

An obvious difference between the high-scoring and low-scoring groups lay in the frequency of the "urge" element. This disparity suggests that students with high scores paid more attention to group discussion progress and actively promoted the problem-solving process, while students with low scores lacked awareness of their task progress, often leading to uneven time allocation and further incomplete discussion.

In addition, obvious differences existed between high-scoring and low-scoring groups in the "evaluate" element of their discussions. Students with high scores evaluated their problem-solving process and results more frequently and could summarise their current task progress, enabling them to identify shortcomings in their problem-solving process and solve the problem better.

The "artifact" element appeared in high-scoring groups' discussions an average of 104 times, compared to 46 times in low-scoring groups, indicating that high-scoring groups were more active in presenting discussion results on task sheets. Although the low-scoring groups drew some valuable conclusions in their discussions, they spent little time writing them down, which led to their low scores.

Overall, there were significant differences between high-scoring and low-scoring groups in several elements, including "respond," "urge," evaluate" and "artifact."

**Fig. 6.2** Process model of students' mathematical communication

**Fig. 6.3** First-level coding results

Figure 6.4 shows the similarities and differences between high-scoring and lowscoring groups in second-level coding results.

**Fig. 6.4** Second-level coding results

There were significant differences between high-scoring and low-scoring groups in the "negotiate" and "quarrel" modules. There were, on average, 15 "negotiate" modules in the high-scoring groups and only three in low-scoring ones; in contrast, the mean number of "quarrel" modules in low-scoring groups reached 33, compared to 13 in high-scoring groups. This indicates that high-scoring groups could often reach a consensus, while low-scoring groups found doing so difficult because rapid topic changes and disruptions prevented thorough discussion of a question.

In total, the high-scoring groups experienced 13 "artifact" modules, while the low-scoring groups experienced nine, in part because low-scoring groups lacked awareness of the need to present discussion results on task sheets. Some low-scoring groups made some achievements in their discussions but did not write down their answers, which had a powerful impact on their final scores.

The "evaluate" module was very important in the problem-solving process. As shown in Fig. 6.4, high-scoring groups experienced six "evaluate" modules, while low-scoring groups only experienced two. Two low-scoring groups did not complete the final "evaluate" module, largely because their problematic time allocation prevented them from completing their discussion within the required time. This indicates that students with high scores paid more attention to summarising and evaluating their results after completing the task, while students with low scores often had neither the time nor consciousness to make a final summary and evaluation. There were also some differences in "analyse the problem," "make a plan," "off-task discussion" and "calculate" modules.

Thus, the mathematical communication characteristics of high-scoring and lowscoring groups can be summarised as follows:

#### (1) **Students in high-scoring groups more actively responded to other students' views**.