Keywords

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 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:

  1. (1)

    What are the elements of students’ mathematical communication in open-ended collaborative problem solving?

  2. (2)

    What are the characteristics of students’ mathematical communication in open-ended collaborative problem solving? Is there any model of students’ mathematical communication?

  3. (3)

    What are the characteristics of mathematical communication between high-scoring and low-scoring student groups in open-ended collaborative problem solving? Is there any difference in the number of communication elements used in high-scoring and low-scoring student groups? Are there any differences in the roles of students’ mathematical communication in high-scoring and low-scoring groups?

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.

Table 6.1 Scoring criteria
Full size table

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.

Fig. 6.1
figure 1

Scoring results

Full size image

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 high-scoring and two low-scoring groups were selected from the two classes taught by Teacher B (in the same school as Teacher A).

Table 6.2 Participations
Full size table

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.

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.

$$\begin{aligned} & A_{1} = \left\{ {E_{1} ,E_{2} , \ldots ,E_{n{1}} } \right\},\;\;A_{2} = \left\{ {E_{n{1} + {1}} ,E_{n{1} + {2}} , \ldots ,E_{n{2}} } \right\}, \\ & A_{3} = \left\{ {E_{n{2} + {1}} ,E_{n{2} + {2}} , \ldots ,E_{n{3}} } \right\} \\ \end{aligned}$$

yielded the union of the above sets.

$$A = \left\{ {E_{1} ,E_{2} , \ldots ,E_n } \right\}.$$

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:

$$A^{\prime} = \left\{ {E_{1} ,E_{2} , \ldots ,E_m } \right\}.$$

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 high-scoring 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.

Table 6.3 Elements of mathematical communication
Full size table

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.

Table 6.4 Elements of mathematical communication (improved)
Full size table

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.

Table 6.5 Precoding results
Full size table

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.

Table 6.6 Elements of mathematical communication (Improved)
Full size table

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 6.7 Modules of mathematical communication
Full size table

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.

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.

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.

Table 6.8 Postcoding results
Full size table

By integrating the analysis of the ten communication modules with Polya’s four-stage 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.

Table 6.9 Three-level coding framework
Full size table
  1. (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 6.10 First-level coding results
Full size table

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.

Table 6.11 Second-level coding results
Full size table

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:

  1. (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 “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.

  1. (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.

  1. (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.

  1. (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.

  1. (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.

  1. (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.

Fig. 6.2
figure 2

Process model of students’ mathematical communication

Full size image
  1. (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.

Fig. 6.3
figure 3

First-level coding results

Full size image

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.”

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

Fig. 6.4
figure 4

Second-level coding results

Full size image

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 low-scoring groups can be summarised as follows:

  1. (1)

    Students in high-scoring groups more actively responded to other students’ views.

    High-scoring groups had far more “respond” elements than low-scoring groups, indicating they were more active in responding to other students’ views. They were not only concerned about their questions and opinions, they were also willing to discuss other students’.

  2. (2)

    Students with high scores paid more attention to group collaboration.

    The high-scoring groups had significantly more dialogues with “urge” and “evaluate” elements than low-scoring groups. This indicates that high-scoring groups not only paid attention to the group’s problem-solving process, they also focused on the group’s collaboration and more actively supported the progress of its discussions.

  3. (3)

    Students with high scores paid more attention to preserving written results.

    High-scoring groups had many more “artifact” elements and modules than low-scoring groups, indicating they paid more attention to the discussion process and to presenting results in written form. They were more active in showing discussion results on task sheets.

    However, there were exceptions. As shown in Table 6.10, Low-scoring Groups 2 and 3 experienced three and four “artifact” modules, respectively. However, after further examination of these two groups’ dialogues, we found their “artifact” modules all appeared at the early stage of their discussions. Low-scoring groups began to fill out their task sheets before reaching an agreement and, therefore, usually needed to spend a significant amount of time modifying or rewriting their answer later.

    In addition, high-scoring groups were better at presenting the results of their discussions on task sheets, actively adopting various methods to optimise their written presentation of task results. For example, in High-scoring Group 1, Student 2 proposed writing drafts in pens and drawing the finalised design in pencil. Student 3 proposed marking all possible rooms first, then ticking off the rooms they chose to use. There were many other proposals and dialogues like these, as students with high scores generally spent more time presenting their results on task sheets.

  1. (4)

    Students with high scores were more likely to reach an agreement in their discussions.

    High-scoring groups had more “negotiate” modules and fewer “quarrel” modules than low-scoring groups. There was no significant difference in the number of “argue” modules between high-scoring and low-scoring groups. This indicates that students in low-scoring groups were willing to question and refute others’ arguments in the mathematical communication process, but had difficulty reaching agreements.

  2. (5)

    High-scoring groups experienced a more complete mathematical communication process.

    Among the four high-scoring groups, two experienced all four aspects of the problem-solving process—analysing the problem, making a plan, practicing the plan and summarising and evaluating. Low-scoring groups’ mathematics communication process was less complete than high-scoring groups’, with none of them completing all four steps. For exampe, Low-scoring Group 4 only experienced one part of problem solving, missing the other three.

  1. (III)

    There were differences in students’ roles in mathematical communication between the high-scoring and low-scoring groups.

    Each student plays a different role in the collaborative problem-solving process and contributes differently to solving the problem.

  1. (1)

    Leaders

We calculated the percentage of each student’s dialogues to examine the different contributions they made, as presented in Table 6.12.

Table 6.12 Percentage of dialogues within the group
Full size table

Table 6.12 shows that the percentage of each student’s dialogues in each group was unbalanced. Students who spoke the most accounted for over 40% of all group dialogues, while students who spoke the least accounted for less than 10%, especially in Low-scoring Group 2, where Student 4 contributed 0% of all dialogues. This phenomenon was common in both high-scoring and low-scoring groups, indicating that dialogue imbalance was not the reason for low scores.

In each group, one or two students had the largest number of dialogues. However, it is not reasonable to determine group leaders based only on the number of dialogues. In High-scoring Group 1, there were eight “assign tasks” and “propose a plan” elements, five of which were put forward by Student 2, for example:

Student 2: No problem, I’ll take the carbon pen first. The carbon pen first, and then the pencil.

Student 2: Draw a draft last. Take a pencil first.

In addition, Students 1 and 2 often played the role of topic proposers in the group discussion, proposing 16 of the group’s 44 “express views” and “pose a question” elements. Students 1 and 2 generally controlled the direction of the discussion and led the group to solve the problem:

Student 1:60 m2. An apartment may have a bedroom, a living room.

Student 1: But the question is how to calculate area?

Student 1: What’s the living room for?

Student 2: Ok, it’s time to write how many square meters each room occupies. The bedroom must be 25 m2, 25.

Student 2: Agreed, and let’s start drawing. We still have to draw the function of the room.

Therefore, this study calculated the percentage of “express views” and “pose a question” elements proposed by each student to decide the leaders of groups. Students who accounted for more than 30% of dialogues in these two elements were considered group leaders (see Table 6.13).

Table 6.13 Percentage of “express views” and “pose a question” elements of each student
Full size table

Based on Table 6.13, the eight groups can be roughly classified into three categories: no-leader groups, single-leader groups and double-leader groups. Low-scoring Group 3 was a no-leader group. High-scoring Group 4, Low-scoring Group 1 and Low-scoring Group 2 were single-leader groups. High-scoring Group 1, High-scoring Group 2, High-scoring Group 3 and Low-scoring Group 4 were double-leader groups.

The proportion of double-leader groups was higher in high-scoring groups. Discourse hegemony did not appear in these groups, meaning every student had the opportunity to speak, and the groups were more likely to have better problem-solving results.

  1. (2)

    Supervisors

Supervisors of groups urged the group discussion’s progress and promoted the problem-solving process. The role of supervisors in groups was identified by analysing the number of “urge” elements in students’ dialogues, as shown in Table 6.14.

Table 6.14 Number of “urge” elements
Full size table

As shown in Table 6.14, there was no obvious supervisor in most groups. Only Student 1 in High-scoring Group 1, Student 2 in High-scoring Group 2 and Student 4 in High-scoring Group 4 showed supervisor characteristics, especially Student 1 in Group 1. There were 31 dialogues with “urge” elements in High-scoring Group 1, 19 of which were from Student 1. For example:

Student 1: Calm down, calm down, now we are going to solve this problem, not biology.

Student 1: Quiet, we’re not here to talk, we’re here to solve the problem, ok?

Student 1: Quiet, don’t worry about that. Let’s start to draw.

Student 1: Okay, shut up, both of you. Enough is enough, enough is enough ah, enough is enough.

This is an important reason why High-scoring Group 1 could successfully complete its task after three unrelated discussions. The supervisor in the group could shut down unrelated topics quickly and focus the discussion on the problem itself.

It is clear from Table 6.14 that high-scoring groups had a higher proportion of supervisors. Supervisors monitored the groups’ problem-solving process and were an important factor in guaranteeing the groups completed their tasks in time. When the direction of the discussion deviated in a no-supervisor group, no student could stop it in time, leaving insufficient time for a complete and thorough discussion.

  1. (3)

    Marginalised students

As can be seen in Table 6.12, marginalised students—such as Student 4 in High-scoring Group 1, Student 4 in High-scoring Group 2, Student 4 in High-scoring Group 3, Student 1 in High-scoring Group 4, Student 4 in Low-scoring Group 1, Student 4 in Low-scoring Group 2, Student 4 in Low-scoring Group 3 and Student 3 in Low-scoring Group 4—provided <15% of the dialogues in both high-scoring and low-scoring groups. Some of these students chose not to participate in the group discussions, while others were excluded and marginalised by other students.

The existence of marginalised students did not affect the discussion results. However, marginalised students will undoubtedly gain much less from the whole collaborative problem-solving process than other students. Teachers should pay more attention to these students by intervening in groups’ collaborative problem solving and encouraging them to participate actively in the mathematical communication process.

  1. (4)

    Blockers

In some groups, some students blocked the progress of group discussion. For example, in Low-scoring Group 3, Student 3 kept talking about his home and starting unrelated discussions, such as:

Student 3: Where do you live?

Student 3: How luxurious my home is, that’s it.

These students blocked the group problem-solving process and led the discussion off-topic in an unrelated direction. Blockers were identified by calculating the proportion of each student’s dialogues that were in the “unrelated discussion” module, as displayed in Table 6.15.

Table 6.15 Percentage of students’ dialogues in “unrelated discussion” module
Full size table

However, the high proportion of some students’ dialogues in the “unrelated discussion” module was probably due to the large total number of their dialogues. Therefore, to locate blockers in groups more accurately, we subtracted the proportion of students’ dialogues from the proportion of students’ dialogues in “unrelated discussion” module. The results are shown in Fig. 6.5.

Fig. 6.5
figure 5

Changes in the percentage of students’ dialogues

Full size image

A positive number indicates a higher proportion of student dialogues in the “off-task discussion” module, and a negative number indicates a lower proportion.

Figure 6.5 suggests that the proportion of dialogues in the “off-task discussion” module was significantly lower for some students, including Student 2 in High-scoring Group 1, Student 4 in High-scoring Group 2, Student 2 in Low-scoring Group 2 and Student 4 in Low-scoring Group 4. In particular, Student 2 in Low-scoring Group 2 and Student 4 in Low-scoring Group 4 scored lower than 20%, indicating they paid more attention to problem-related topics and seldom participated in off-task discussions.

However, blockers were also found in these groups, such as Students 1 and 3 in Low-scoring Group 2 and Students 1 and 2 in Low-scoring Group 4. Blockers were mainly found in low-scoring groups, as they blocked the progress of group discussion and made it difficult for the group to focus on the problem.

6.4 Discussion and Conclusion

  1. (I)

    Discussion

  1. (1)

    Respond actively to other students’ views

Constructivist learning theory believes that knowledge construction occurs in the interaction between learners; thus, it places great emphasis on “conversation” and “collaboration” (Chen & Zhao, 2012). Li (2001) pointed out that from this perspective, the mathematical communication between students should be active, and students should fully participate in the discussion. She further emphasised that understanding of problems is gradually deepened through the process of students’ questioning, responding, reflecting and generalising. Thus, a positive response is of great significance in the mathematical communication process. Students promote the problem-solving process through questioning and responding to others’ views, discussing the problem in great depth, and solving the problem.

Comparing the “respond” element between high-scoring and low-scoring groups reveals that high-scoring groups responded to others’ opinions, questions, doubts and refutations more frequently and positively. In contrast, low-scoring groups were more likely to give no response to others. For example, in Low-scoring Group 2:

Student 3: Or there is no room here.

Student 2: You can draw a balcony and an open-style kitchen.

Student 3: Draw another bedroom.

Student 1: Let’s draw a restaurant.

Student 3: It’s impossible that he has a wife and children.

Student 2: Why is your door different from the others?

Student 2: Have you calculated how many square meters these rooms can have together? This is 6, it’s only 6 m2, 6 times 10, it’s only 6 m, you draw 3 and 2 m.

Student 3: Forget it, I quit, you can draw it.

Student 1: What is this? You know, there’s no problem if we expand it.

In this dialogue, Students 1–3 put forward many views and questions (draw a balcony, open-style kitchen, bedroom, etc.). However, these views and questions did not receive positive feedback, making the discussion topic shift too fast. The discussion of the room’s function had not yet ended when the topic shifted to the room’s size.

Therefore, students should pay attention to others’ views and questions in the discussion process and respond actively to ensure the group discussion remains focused on one topic. In the intervention of students’ mathematical communication, teachers should not only pay attention to the results of problem solving, but also to students’ discussions and remind them to respond to others’ opinions.

  1. (2)

    Communication consistency

Coding students’ mathematical communication revealed that “negotiate,” “argue,” and “quarrel” were the basic communication modules of collaborative problem-solving process. They were also the three critical communication modules in promoting the problem-solving process. Therefore, students should pay full attention to these three communication modules.

There were obvious differences in “negotiate” and “quarrel” modules between high-scoring and low-scoring groups. Whether a group can reach an agreement during its discussion is particularly important in the problem-solving process.

Previous research found that after making some aspects of tasks public, there would more likely be a shift to cognitive authority (Langer-Osuna et al., 2020). However, in the current research, when discussing a given issue, students commonly had different opinions and questioned each other. This cognitive conflict is constructive in open-ended collaborative problem solving. However, if students ignored this conflict and started to discuss other issues, they would need to discuss this topic again later. This will waste time and affect the atmosphere of group communication. Therefore, students should pay attention to communication consistency.

In the intervention process, teachers should pay full attention to conflicts among students and encourage students to resolve them through group discussion. However, if teachers find that students have failed to resolve the conflicts despite sufficient discussion, they should intervene (Dong et al., 2013).

  1. (3)

    Presentation of results

The “artifact” module is often overlooked in the collaborative problem-solving process. However, the differences between high-scoring and low-scoring groups showed the “artifact” module’s indispensable role in the problem-solving process. After completing their discussion, students should present the discussion results on the task sheets in detail. Especially in open-ended collaborative problem solving, teachers and students should pay more attention to the answer’s completeness. In the task given by this research, students could also do extra exploration in addition to the basic requirements, such as providing a variety of house designs, describing the situation in detail and so on.

However, writing down answers at the beginning of group discussions without reaching an agreement is inadvisable, as students will often need to make numerous modifications to these answers during the discussion process, thus wasting time.

  1. (4)

    Integrity of the mathematical communication process

Polya (1945) divided the problem-solving process into four stages: “Understand the problem,” “Devise a plan,” “Carry out the plan” and “Look back.” He stressed that each stage is important and cannot be replaced.

The coding results for the students’ mathematical communication models suggest that students and teachers did not do enough to “analyse the problem,” “make a plan” and “evaluate” parts of the mathematical communication process. Teachers should let students go through the complete collaborative problem-solving process to ensure the completeness of the mathematical communication process. Students should start with practical problems and then summarise the mathematical problems through analysis. Teachers should not do this for the students but may give them some hints. In addition, teachers should remind students to make a plan to solve the problem, rather than directly starting to discuss some of its details. Teachers could also organise students to report or adopt other means to summarise and evaluate the problem-solving process and results. Students could also independently carry out the summary and evaluation work in the mathematical communication process, which is an important module for helping students find mistakes and adjust the problem-solving direction. Students get the most from problem solving and develop their mathematical communication ability best when they undergo the whole mathematical communication process.

  1. (5)

    Students’ participation in mathematical communication

Through observation, Wang (2002a, 2002b) found that imbalance is very common in students’ participation in collaborative learning. In this case, students with higher participation levels tended to gain more than those with lower participation levels (Wang, 2002a, 2002b). This study also found a similar phenomenon by analysing students’ contributions to dialogues. In both high-scoring and low-scoring groups, marginalised students contributed to only a small number of dialogues. While this did not significantly impact the groups’ problem-solving results, the marginalised students will undoubtedly gain less from the collaborative problem-solving process than other students. Therefore, teachers should pay special attention to these students during an intervention, analyse why they are marginalised and encourage them to participate actively in group discussions.

Our review of group discussion content revealed that some teachers paid attention to marginalised students in some groups. For example, in High-scoring Group 4, the teacher tried to let Students 1 and 3 participate in the discussion between Students 2 and 4, as shown below:

Teacher: Did you split up again?

Student 2: Based on Feng Shui, the house must be square.

Teacher: I want four of you to discuss, you are now stuck at two, right?

Student 2: And you can’t slam the door.

Student 2: What do you think if we put a toilet next to the bedroom?

Student 4: If theirs doesn’t work, try ours.

Student 2: Let’s combine the bathroom with the bedroom, and the dining room, all in a small space.

However, it can be seen from the above dialogue that students did not listen to the teacher’s arrangement. Students 2 and 4 continued their discussion, and Students 1 and 3 were not involved and eventually asked their peers, “Are you unhappy to have stupid teammates like us?” In the after-class interview, the teacher revealed that the four students in this group were not partners in regular math classes and seldom discussed issues; thus, in this lesson, these students might not have known each other well and may even have hated each other. This is an important reason why this group had difficulty conducting mathematical communication. This situation suggests that a long time is needed to form groups for collaborative mathematics problem solving.

Although some studies and theories support that heterogeneous groups perform better than homogeneous groups in collaborative learning (Bowers et al., 2000; Wang, 2002a, 2002b), Wang and Chen (2008)’s experimental study found that, although heterogeneous groups are more suitable for solving open-ended collaborative problems, the differences between heterogeneous groups will also lead to communication barriers that hinder the problem-solving process. It has not yet been agreed upon whether heterogeneous or homogeneous groups are more conducive to students’ collaborative problem solving.

Therefore, while teachers can consider the principle of “homogeneity between groups and heterogeneity within groups” when grouping students, they should also fully consider students’ personality characteristics. Teachers should not assign students with conflicts to the same group and should not regroup existing regular in-class discussion groups without considering students’ opinions.

  1. (6)

    Students’ roles

There were significant differences in students’ roles in the problem-solving process between high-scoring groups and low-scoring groups, especially for supervisors. Although teachers play a supervisory role in the collaborative problem-solving process, it is difficult for teachers to pay attention to every group’s discussion or monitor their problem-solving process. There are usually five to eight groups in every mathematics class in China. Teachers could set up supervisors within groups who would be mainly responsible for the following group tasks: (1) supervising the problem-solving process and reminding the group it is behind schedule; (2) stopping group members from discussing topics unrelated to the problem and maintaining the group’s discussion discipline; (3) encouraging marginalised students to participate more in group discussion; and (4) maintaining a good group discussion atmosphere, identifying problems that cannot be solved through group discussion and seeking help from teachers when groups have conflicts and cannot reach an agreement.

  1. (II)

    Conclusion

This study has constructed a set of elements of students’ mathematical communication in collaborative problem solving and established a three-level coding framework. By coding the mathematical communication of eight selected groups, we have summarised secondary school students’ mathematical communication characteristics.

Students promoted the problem-solving process through “negotiate,” “argue” and “quarrel” modules and presented their discussion results on task sheets in “artifact” modules. While students were not afraid to express their opinions and refute other students’ opinions, there were various problems in the mathematical communication process, including having difficulty reaching agreement, an incomplete mathematical communication process, etc.

We compared the differences in high-scoring and low-scoring groups’ mathematical communication characteristics from two aspects—the number of communication elements and students’ roles in mathematical communication. The study results showed that high-scoring groups had advantages in various aspects. High-scoring groups were more active in responding others’ views, reached agreement more frequently, were better at presenting results on task sheets, followed a more complete collaborative problem-solving process and paid more attention to the progress of group discussion. There also existed differences in students’ roles between high-scoring and low-scoring groups. For example, supervisors were more common in high-scoring groups, and blockers were more common in low-scoring groups.

  1. (III)

    Inspiration

Since the new curriculum reform, the “student-centred” principle has entered Chinese mathematics classes, and students’ participation in mathematics classes is becoming wider and deeper (Zhao et al., 2019), while the international situation is similar (Wang et al., 2013). Collaboration is receiving increasing attention in mathematical problem solving research (Liljedahl & Cai, 2021), and collaborative problem solving has also attracted extensive attention from researchers and teachers. Our analysis of mathematical communication in collaborative problem solving revealed that students are not adaptable to this teaching method. This can mainly be seen in the incomplete process of students’ collaborative problem solving; students do not know what to do when given an open-ended collaborative problem-solving task.

In addition, students also frequently asked teachers for help during the collaborative problem-solving process, asking questions (e.g., “Does the balcony count as a room?” “Does the toilet count as a room?” “Does the corridor count in the total area of the room?”) to which there was no specific answer. This shows that students were more accustomed to traditional teacher-led teaching than student-led teaching based on problem solving. Even though student-centred instruction has been widely proven more effective than teacher-centred instruction (Granger et al., 2012), the teacher-centred instruction has enjoyed prominence for decades in mathematics education (Stephan, 2020) which possibly set barriers for students’ collaboration on the open-ended tasks. Students tended to ask teachers for help rather than discuss issues with their peers. Restricted by classroom time limits and heavy teaching loads, it is difficult for teachers to incorporate problem solving into their daily classroom teaching. Consequently, students have long been unable to develop their collaborative problem-solving and mathematical communication abilities.

Teachers could explore integrating problem solving into their daily teaching further. For example, in some parts of a lesson (e.g., introducing new lessons or exercises), teachers could let students discuss or debate issues to improve their collaborative problem-solving and mathematical communication skills.