Research on Student Interaction in Peer Collaborative Problem Solving in Mathematics

Abstract

Collaborative learning based on mathematical problem solving is an important way of organising teacher teaching and student learning. Two-peer collaboration is more common in classrooms. This study explored the characteristics of and differences in the interactive performance of high and low academic performance students by analysing the collaborative mathematical problem solving processes of heterogeneous peer students in seventh grade. Nine peer groups with the highest academic gap among 54 are purposefully selected for the current research. The study found that high academic performance students produced a significantly higher amount of discourse than low academic performance students, and that there were significant differences in indicators such as order, negation, and explanation. When the gap between the two peers’ academic performance decreased, peer-to-peer communication was more equal, and low academic performance students actively participated in the problem solving and gradually fought for task initiative.

8.1 Introduction

The consensus in the academic community is that collaborative learning plays a critical role in promoting students’ learning and building social relations among students. Therefore, analysing students’ performance in collaborative problem solving is of great significance for understanding students’ collaborative process and improving teachers’ understanding of collaborative problem solving. In current practice, the most common way of collaboration in primary and secondary schools is collaboration between two students. Therefore, this article will focus on the interaction of two-student collaboration pairs to analyse their interactions during collaborative problems solving (CPS) in mathematics. The study aims to gain insights into the performance of peer collaboration in the CPS process and provide implications for teachers for instruction when teaching CPS in mathematics.

Although many studies pointed out that collaborative learning had a positive effect on developing students’ abilities, the internal structure of multi-person groups had some drawbacks. On the one hand, some students did not actively participate in collaborative discussions; on the other hand, it was difficult for teachers to pay attention to each student’s performance, as doing so might cause some distractions from the given tasks (Sun & Wen, 2004). Zuo and Huang (2010) concluded that it was easier to build a consciousness to collaborate between pairs and that their independence could be stronger than in multi-person groups. Fleming and Alexander (2001) set up peer-collaboration treatment conditions and individual tasks (control conditions) to investigate whether the observed benefits of peer collaboration lasted for a certain time. Their study showed that students in the treatment condition did more outstanding performance than those in the control condition in strategy use, metacognitive understanding of strategy chosen, and recall gain. Compared with independent learning, peer collaborative learning could help to develop students’ creative problem posing (Han et al., 2013), which further affected their ability to solve mathematical problems (Zhang et al., 2019).

One issue concerning how teachers organise collaborative groups in teaching practice is how to categorise students into groups or pairs, usually referring to whether groups or pairs should include students with similar or different traits. Students in pairs can have different characteristics, and different combinations of students may influence their collaborations. Students’ ability levels can be a key factor affecting their interaction during collaborations to complete mathematical tasks (He, 2005). Most researchers favoured heterogeneous groups, but others disputed their effectiveness.

One aspect that some studies have confirmed the effectiveness of heterogeneous grouping in promoting students’ abilities. Fantuzzo et al. (1992) paired 80 students with learning difficulties with students with higher academic performance to help the former improve their academic performance. Although the scholar did not explicitly mention that the groups’ heterogeneous structure helped improve the students’ academic performance, the matching method advocated heterogeneous groups or pairs. Zhang et al. (2019) matched 72 seventh grade students in pairs according to their final math scores and gave them a pencil-and-paper test, and found that peer collaboration with heterogeneous structure had the greatest impact on the students’ ability to solve mathematical problems. Other researchers amplified the effectiveness of heterogeneous group structure from a theoretical perspective. From the perspective of group socialisation theory, one of the most effective group structures was heterogeneous (He, 2005). The classic “zone of proximal development” concept pays more attention to the differences in peers’ abilities. It holds that teachers can only promote growth by pairing students with partners with stronger abilities. The learning condition is that the two peers’ ability performance is inconsistent, because their different abilities afford them different views on problem solving. With collaboration, peers reached a consensus through interaction and reflection to make progress (Sun & Wang, 2009). Within the heterogeneous group, the authority distributions vary betweem students (Langer-Osuna et al., 2020). In the heterogeneous group, the high ability students are not affected by the grouping; i.e., the high ability students’ academic performance was relatively independent of the group structure, while low ability students can learn more knowledge in a heterogeneous group (Saleh, 2005).

The other aspect that some researchers are uncertain about the effectiveness of heterogeneous groups. Mortweet and Utley (1999) reviewed a learning model, also called the whole-class paired learning model, one of the most in-depth peer collaborative learning research methods. The model is based on a peer matching form with little difference in ability, so students can get more effective feedback from each other. In this model, most students’ scores could increase between 20 and 70% (Delquadri et al., 1986). Wang and Chen (2008) divided collaborative groups into homogeneous and heterogeneous groups from a cognitive style perspective, and tested them to research the differences in their problem-solving levels. They found that individual differences in heterogeneous groups would become obstacles to effective communication. Bowers et al. (2000) made a meta-analysis of 57 effective duos in collaborative groups to determine whether there was a difference in performance between the heterogeneous and homogeneous groups. The study found that although single studies had shown that heterogeneous groups perform better than homogeneous groups, the overall differences between the two were not significant. Some studies also set up reading tasks for collaborative learning and found that gifted students in homogeneous groups improved more than those in heterogeneous groups (Melser, 1999).

In addition to the above, there is a dispute regarding the structure of collaborative groups. For one thing, an individual’s performance depends on the type of task, so it is difficult to explore which kind of collaboration is better without considering the task type (Bowers et al., 2000). For another, many studies judge the effectiveness of heterogeneous groups based on final test results, so different test methods and tasks may lead to fluctuations in the results. Research based on process performance within a group may result in higher stability than research based on test results. Moreover, there is less research on interaction performance within heterogeneous groups, and more on whole-group rather than individual performance, so it is difficult to see the specific performance characteristics of each student in heterogeneous groups. This study focuses on heterogeneous groups and explores the performance of each student in peer collaboration groups in mathematical problem-solving to determine whether students benefit, which is of great significance for helping teachers organise collaborative groups.

To sum up, pair collaborative problem solving is currently the most widely-used and effective form of collaboration. However, as there are differences in students’ academic performance, we inevitably face the choice of group structure. This paper researches junior high school students’ interaction performance in peer collaboration groups and investigates the efficiency and quality of their interaction processes and characteristics in collaborative mathematical problem solving. There are two main research questions.

• 1. How do students in heterogeneous pairs perform differently in peer interactions in collaborative mathematics problem solving?

• 2. What differences are there in peer interactions among students in heterogeneous peer collaboration groups, based on collaborative mathematics problem solving?

8.2 Research Method

8.2.1 Participants

In the current research, purposeful sampling is selected for investgating the situation of heterogeneous groups. The influence of students’ academic performance on collaborative effect is obvious, and most researchers divide group members based thereon. Meij (1988) found that academic performance affected the number and type of questions students asked their partners. In researching students’ interaction in pair collaboration, two concepts seem particularly useful: student interaction and pair collaboration. Student interaction refers to reciprocal communications, either verbal or nonverbal, among students. Pair collaboration refers to two students sitting together to complete a task through interaction. The research uses “Xiao Ming’s apartment” as the task material for pair groups to solve and selects videos from the problem-solving processes of 106 seventh-grade students (54 peers) from three classes in a district of Beijing. This research defines heterogeneous groups by students’ academic divergence within a pair group. Students’ mid-term test scores (unified district examination, consistent scoring criteria) are taken as their academic performance. The gap in academic scores between the two pair members is arranged in descending order; students in the paired collaborative group with relatively high scores are defined as high academic performance students (HS), while those with relatively low scores are defined as low academic performance students (LS). To ensure that two peers could complete the task, each HS had to score at least 80 of a possible 100 points. The researcher selected the nine paired collaborative groups with most academic gap as research objects. Among them, the terms high and low academic performance are relative and do not refer to an absolute level of performance.

Because collaborative groups are generally two-student groups in daily teaching, and teachers cannot notice and guide each group, pair collaboration was carried out without teacher intervention in this study; teachers did not guide students on how to complete the task. The choice of research samples was in line with our daily teaching situation.

8.2.2 Encoding Frame

To deepen the research, researchers have gradually shifted attention from the effect of collaborative learning and influencing factors to the collaborative process. In the pair collaboration process, interaction cannot be avoided, and the interaction between two peers can help us understand and analyse students’ interactive performance in the collaboration process (Wang, 2004).

Saleh et al. (2007), noting that students with general ability often did not make full use of heterogeneous group learning, proposed structured collaboration to help students overcome this participatory inequality. His study divided verbal interaction into 11 indicators: statement, argument, evaluation, question, request, proposal, confirmation, negation, repetition, order, and off-task. The scholar’s coding of verbal interaction is more detailed, covering most dialogue behaviours and clearly describing each index.

As Saleh’s coding framework marks most utterances as “argument,” this study combined it with Gillies’ (2003) and Guiller’s et al. (2008) frameworks to encode the interaction process in pair collaboration more carefully. Saleh’s “argument” dialogue in the coding framework was divided into two categories: “giving a point of view” and “explanation,” and the indicator of “checking” was added. After generating the draft coding framework, part of the data is coded to revise the framework. However, it was found that the “checking” dialogue did not appear in each group, so this category was deleted. Because the research object of this study is paired collaborative group heterogeneous paired collaborative groups, the academic performances of the two students in the group are quite different. Watching the video revealed that they often did not communicate; therefore, the researcher added the index “no communication” to the coding frame to further the research. In this study, four students shared a large desk, with adjacent students acting as a paired collaborative group to complete peer tasks. As each paired collaborative group’s interactions could be subject to external influence, the “out of group discussion” index was added to the coding framework. The final coding framework was based on this study’s research content, complemented by multiple viewings and analyses of specific videos by the researcher, as shown in the following Table 8.1. As the framework is conducted by previous research, the validity is ensured. After coding, the researcher conduct a re-code to ensure the reliability of the coding result.

Table 8.1 Coding framework

8.3 Results

Per the research scheme, this section further explained and refined the research process, and classified and compared the paired collaborative groups based on their interaction performance. The change in task set location was an odd aspect of this study. Students’ verbal interactions were first analysed, followed by the relationship between task set location and the two peers’ academic performance gap.

In this study, NVivo12 was used to split the coding of nine pair groups of videos and calculate the length ratio of each interaction category. The coding results were analysed in R3.6.3 and SPSLS6.

8.3.1 Verbal Interaction

In this study, the video was analysed as a continuous sample, and discourse meaning was used as a coding unit to count the proportion of each interaction category rather than just recording utterance frequency, for more accurate analysis results. This study encoded and analysed videos of nine pair groups, counted the proportion and total proportion of HS and LS in 14 behaviours (13 verbal interaction and one “no interaction” categories), and calculated the corresponding average values, as shown in the following Table 8.2.

Table 8.2 Overall interaction proportion between HS and LS

As shown in the Table 8.2, the high academic performance students’ proportion of verbal interaction time (47.36%) was significantly higher than that of low academic performance students (22.28%), to some extent indicating that LS could not fully participate in the discussion, while HS dominated the conversation. In addition, “no communication” accounted for 30.31%, close to one-third of the total task duration, which reflects that communication in the two students’ interactions was slightly less active and unequal in the nine pair groups.

The Table 8.2 shows a significant difference in the duration proportion of each indicator between HS and LS. To further illustrate this point, the researchers made a two-factor repeated measurement analysis of variance on the proportion of the two students’ dialogues corresponding to the nine video groups for each indicator. The difference in the data corresponding to the dialogue categories was statistically significant (p < 0.01).

Fig. 8.1

Two-factor repeated ANOVA results

The two-factor repeated analysis of variance results (Fig. 8.1) show significant differences between HS and LS in repetition, evaluation, explanation, order, and negation. The researchers focused on the data for each indicator to further understand the conversational differences between peers in the collaboration process. “Evaluation,” “explanation,” “giving a point of view,” “order,” and “negation” of these dialogues were significantly higher in HS than in LS. Dialogue representation with critical meaning, such as “evaluation,” “order,” and “negation,” was higher in HS than in LS; LS's utterance proportion for these three indicators was close to 0, similar to the results of Saleh’s study of students’ verbal interaction (Saleh et al., 2007). In addition, LS were higher than HS in repetition and suggestion.

Certain differences were found in the dialogue duration between groups. Systematic clustering of the data on the two students’ dialogue proportion duration corresponding to each video analysis result divided the nine groups into two categories, with the first to the fourth groups in one category and the remaining five in the other. The box diagrams in the following Fig. 8.2 clearly depict this classification difference in the two groups’ conversation duration data.

Fig. 8.2

Box plot of group dialogue duration

A is the first to fourth groups and B is the fifth to ninth groups. There is a clear stratification in the dialogue duration between the two groups, with A’s dialogue duration being lower than B’s. The gap in A’s two-student midterm test scores was over 40 points, while B’s was between 10 and 40. Based on this finding, the following group analysis is divided into two categories, Category A and Category B.

8.3.1.1 Verbal Interaction of Category A

There were large gaps between the two students in each Category A. The average dialogue duration in the task-solving process was 48%, meaning over half of each group’s time passed in a state of non-communication. The proportion and total proportion of HS and LS for the 14 indicators are shown in Table 8.3.

Table 8.3 Interaction between the HS and LS in category A

HS and LS showed more significant differences in dialogue categories in Category A. As in the previous part, HS were significantly higher than those with LS in terms of “order,” “negation,” “explanation,” and other indicators. The differences in dialogue between HS and LS in Category A exceeded the nine groups’ average performance. Even in the “out of group discussion” indicator, HS in Category A showed a significantly increased proportion, indicating more communication and discussion with non-group members.

While the overall communication time between the two students was less, HS spoke longer than LS, and the LS expressed fewer opinions. LS lacked confidence and dared not express themselves, were suppressed or ignored by HS, and were not recognised by their peers, resulting in less interaction. This phenomenon was an inherent risk when using heterogeneous learning teams (Slavin, 1991).

[Episode 1]

LS: Is it possible that he has one student, father and mother, grandparents?

HS: Don’t look at them, we can count it ourselves.

[Episode 2]

LS: In case, it is impossible for his grandfather and grandmother to have more than fifty, or more than sixty, or more than seventy, plus more than 140, more than 130, and more than 120, but he has only one.

HS: Come on, let’s be 13, let’s be 13.

LS: Figure out four possibilities. show him.

HS: does not respond to this, the two are in a state of non-communication.

These two small peer interaction episodes are from Group 1. In the first episode, LS cast an idea, but HS ignored it and did not respond positively. Similarly, in the second episode, LS analysed the problem, but HS’s response was not related to LS’s analysis and instead directly advanced their views. At this time, LS did not adhere to their views and compromised, suggesting that four possibilities should be calculated to show the teacher; HS did not respond to this, and the communication between the two was interrupted. Analysis of the two episodes showthat neither communication formed a complete collaboration process, and LS could not get a response of feedback after expressing their views. A complete collaboration should include an interactive process, expression, listening, questioning, etc. These behaviours collide in collaboration and communication to promote the problem’s solution (Cao & Bai, 2018). However, the above episodes had no interactive process, leading to the collaboration’s failure.

[Episode 3]

HS: If 30, 30–10 is 20. 32, let’s get it 32.

No response.

HS: 32 + 32 + 12 = 

No response.

HS: 64, 64, 76

No response.

HS: Well, it’s 25. If it happens to be 25, his brother and sister are 25 years old. Brother, then it’s 13. he was 13 years old.

No response.

HS: 13 * 2, 26.

No response.

HS: Then make them 15 years old!

HS: If they were 15, 15, one more, 15

Episode 3 is a dialogue episode from Group 4, located in the middle and late stages of the whole discussion process. No response meaned no one spoke at all during this period. In this episode, HS constantly expressed his opinion, but LS did not speak a word and had no response to HS’s utterances. This completely non-responsive state (often found in Category As) was not related to HS’s interruptions or disregard but to LS’s failure to take the initiative to respond.

This may have been because LS was aware of their inability to participate in the collaboration and consciously gave up on the task. In other words, they were not listening to HS. It is also possible that LS’s academic performance could not reach a certain level, making it difficult for them to express their ideas even though they were listening carefully to the HS. No matter the reason, the episode showed that a large enough gap between the two students’ scores leads to a gap in their exchanges, making it difficult for each to supplement the other’s views and preventing them from reaching the “thinking collision” state needed to promote constant problem solving.

[Episode 4]

LS: is it all equal to this?

HS: you don’t move!

…

LS: poop

HS: tell you, don’t mess up!

LS: hum

HS: I didn’t want to be in a group with you. Don’t mess up.

On the one hand, Episode 4 responded to the above-mentioned differences in dialogue categories between HS and LS. On the other hand, it also showed that HS resisted LS when they expressed their opinions or took some actions, preventing in-depth collaboration. To a certain extent, this situation explained why the dialogue duration between the two students was less than half of the total task time—they could not collaborate or communicate peer-to-peer, rarely exchanged views, and seldom thought deeply, eventually resulting in HS solving the problem by themselves. It was not an ideal peer collaboration.

In peer collaboration based on mathematical problem solving, a large gap between the two partners’ ability to understand and solve problems makes their effective collaboration difficult. The two have few dialogues on tasks. LS were likely to be excluded or dominated by HS, and lacked the confidence to express their opinions. In other words, individuals in collaborative groups need to be recognised by other members. When the performance gap between the two is large enough, it is difficult for LS to obtain sufficient HS identity, leading to their marginalisation in or even separation from the collaborative peer group.

8.3.1.2 Verbal Interaction of Category B

The academic performance gap between the two students in each Category B was narrower, and the average group dialogue duration was 86.67%, much higher than in Category As. The communication between the peers was in a positive dialogue state. This study counted the proportion and total proportion of HS and LS for the 13 specific dialogue categories, and the average proportion of non-communicative behaviour in Category Bs, as shown in the Table 8.4.

Table 8.4 Interaction between the HS and LS in Category B

Category B HS’s discourse proportion was 56.05%, while LS’s was 30.61%, both higher than in Category As. In addition, the differences in dialogue between HS and LS were significantly reduced, particularly in the indicator of “explanation,” as were the differences in other indicators.

The Fig. 8.3 shows the performance of HS in Category A and Bs in various dialogues categories. A- and B-group HS’s performances were largely consistent for such indicators as “statement,” “repetition,” and “request,” while there were differences in Category A-HS’s performance in such indicators as “explanation,” “proposal,” “order,” “negation,” “off-task,” and “out of group discussion.” HS’s proportions of “order,” “negation,” and other indicators with strong words were significantly reduced, showing that HS recognised LS’s peer role to a certain extent, regarded it as the object of equal communication, and were more willing to explain their ideas to them. The communication state between the peers was more positive, more coherent, and smoother, complementing each other. The LS gradually began to fight for the right to speak for themselves in the discussion process, which was no longer HS’s solo play.

[Episode 1]

LS: let’s reason

HS: her parents must have…

LS: Mom and dad must be over 70

HS: Mom and Dad, oh, this is the sum, the sum

LS: almost 70 years old, all in all

HS: that’s for sure, or a person is 70 years old, ha-ha-ha

LS: her parents must be over 70

HS: but, 42

LS: a person is 35. for example, over 70 years old, the average age is 70,

HS: she has a big sister and a little brother

Fig. 8.3

The duration of dialogue categories of HS in A&B

The LS in the above episode constantly gave opinions, leading the dialogue in this small segment. At the same time, the HS was constantly echoing and thinking. The two peers were giving each other ideas and opinions so the direction of the discussion can be carried out smoothly. This kind of dialogue mode was much better than that in Category As; the two students complemented each other’s views, communicated as equals, and expressed their views and thoughts to a greater extent.

[Episode 2]

LS: one of them is 28 years old

HS: how do you know?

LS: there is a middle age. You can either add one at the front or subtract one at the back

HS: maybe none of them is 28 years old. The average family is 28 years old,

LS: it’s possible.

HS: they are all 28 years old, and there are four of them, 31, all of them…

LS: you see, you’ve got it. You minus 13 years old, that’s 112

HS: then I want four numbers, no 28. Then you… What should you do then?

LS: wait a minute. I see. You take 112 minus 28, you calculate it.

HS: why do you do with 112 minus 28?

LS: as soon as you subtract 28, you will find the age of the second person, the first

HS: isn’t that the age of the second person

LS: maybe it is!

HS: minus 28, divided by 3, equals 28

LS: why? You can also subtract 28 first

HS: minus 28, and then?

LS: subtract 28 and divide by 3

HS: it’s still 28

In Episode 2, after the two students agreed that the age of the seventh-grade student was 13 years old and the cumulative age of the remaining four students was 112 years, they discussed whether the average age of the four students was 28 years old. Their discussion was in a relatively positive state throughout. The LS constantly raised questions. The HS’s performance differed from Category A-HS’s, offering a positive response instead of negating or interrupting the LS’s questions and opinions. The two students constantly gave opinions and digest each other’s opinions until a consistent conclusion is reached.

However, there are also problems in the dialogue between the peers because the process concerned discussing the students’ average age, and the LS did not understand the concept deeply enough. If one was 28 years old, the average age of the remaining three must also be 28 years old, but the LS did not realise this key point and needed to discuss it with the HS. This discussion had two effects. On the one hand, LS’s views were inefficient and did not promote problem solving; the HS had no problem understanding the average age but did not understand the LS’s purpose, and so always followed their ideas, leading to low efficiency. On the other hand, it greatly promoted the LS’s knowledge understanding because the LS guided the discussion, and the peers constantly exchanged views through equal communication.

[Episode 3]

HS: 28 years old. What if they have a student younger than the seventh grader?

LS: definitely, maybe mom and Dad, and then…

HS: it’s impossible, impossible

LS: even up to mom and Dad, there are brothers and sisters

HS: two people, two

LS: Mom and Dad, brother and sister

No communication

LS: at the age of 28. the seventh grader. 112. Besides the seventh grader, there are four, four.

HS: then, if you look at it, we’ll do it by hand

HS: so, what is 125 minus 13? it is 112, 112. If Mom. If Dad is 38 and mom is 35, it will be over 17. 28 minus 17 is 11.

LS: 11 years old, that’s impossible. He cannot…

HS: there are only four people, and who is a 28 years old relative? Then if this is a second child, this second child will be eight years old and his father will be 40. Dad, mom.

Episode 3 is a group discussion after agreeing that the average age of the remaining four was 28. In this clip, the interaction between the peers is mutual. They first consider that the five people were a family, and then discussed the possible ages and identitied of the family members. When the discussion was is interrupted, the HS offered an idea, suggesing that the father was 38 years old and the mother was 35 years old. According to the average characteristics, 28 should be subtracted from the father’s and mother’s average ages to get the other member’s age. This idea shows that the HS had a good understanding of the concept, was flexible, and constantly pushed the task forward, while the LS constantly collaborated and gave ideas.

While the LS helped identify the five households as one family, the HS promoted the age of each household, because the LS might have difficulty calculating the average but can provide other ideas. This shows that academic performance impacted students’ mathematical problem solving to a certain extent. The communication between the peers shows mutual respect and equality. They threw ideas at each other and absorbed feedback to achieve the purpose of collaboration; it was not a one-person show.

In Episodes 1–3, the performance of Category B-LS differs from that of Category A-LS. The former are more active, have more dialogue behaviours, express themselves actively, and provide feedback to HS. The line chart below compares the discourse performance of LS in Category A and Bs (Fig. 8.4).

Fig. 8.4

The duration of dialogue categories of LS in A&B

In Category A and B, the performance of the LS is obviously different. Firstly, the frequency with which LS actively gave opinions was significantly higher in Category Bs, even slightly higher than that of HS in some groups. Additionally, the duration of their “explanation” and “giving a point of view” utterances was significantly longer. Secondly, both LS and HS used words of “order” in Category Bs, and so were more able to stick to their own opinions, consistent with the above episode. LS in Category Bs were no longer “marginal people” but participated in the collaboration and contributed to the pair collaboration in solving problems.

Based on the above analysis, students’ interactions will be more active and equal when solving mathematical problems in paired collaborative groups if the performance gap between the peers is reasonable.

Category B-HS had different attitudes toward their peers than Category A-HS and communicated more with others in the same position. Category B-HS listened to their LS peers’ opinions and gave appropriate feedback. Correspondingly, LS made great progress, actively expressing their views rather than passively receiving information in a negative position.

8.3.2 Task Set Location

As Category A-LS performance differed from Category B-LS performance, this study generated performance statistics on LS in CPS. The results showed that Category B-LS’s participation degree was high, but Category A-LS’s performance was inconsistent. The HS in Groups 1, 3, and 4 had significantly weaker participation than the HS in Group 2, leading the researcher to review the interactions in Group 2 carefully. The following two episodes show Group 2’s dialogue.

[Episode 1]

HS: did you see the task?… Look the task again.

LS: five households in total.

HS: well.

LS: the average age of them is 25.

HS: well.

LS: one of them is a seventh grader.

HS: well

reflection

HS: how old is the seventh grade?

LS: 12 years old.

[Episode 2]

HS: change these two.

LS: can it be like this

HS: don’t move!

HS: 38, this is 18. It’s easy to give identity to the four people. The younger one is the aunt, this is his mother, this is his brother, this is his little brother.

LS: what about his father?

HS: cannot his father die?

LS: do you think his father cannot go on a business trip?

HS: come on, make up a paragraph

Episode 1 is a short interaction at the beginning of the task. The HS asks the LS to read the task again and approves what the LS said, ensuring they both (especially the LS) understand the task before starting the discussion. Group 2’s HS behad differently than the HS in the other three groups, giving the LS time and opportunity to reflect and speak. However, the HS still dominated the interaction between the peers, who were not completely equal. In fragment 2, the LS was interrupted when trying to share views and denied by the HS, who showed impatience. Although the LS participated in the problem solving and interaction to a greater extent in Group 2 than in other groups, the interaction between the peers was not always active and equal, and there were some problems.

The position of the task set in Group 2 also differed. When solving problems, the task set was mostly placed between the peers in Group 2, while it was almost always placed on the HS’s desk in the other three groups.

Based on the video of the nine groups’ pair collaboration, the following conjectures are offered: when the task set was placed between two peers, the LS was more likely to participate in the interaction; when the task set was placed on the HS’s desk, the LS was more likely to be dissociated or participate in other groups’ discussions.

The researcher analysed that when the task set was placed in the HS’s position, the LS could not get task-related text information and sometimes did not take the initiative to participate. In cases with no interaction, the LS could not get task-related information (text or verbal) and were less likely to participate in the task.

This study counted the interaction duration in the nine groups and found task set location may impact pair collaboration. The task set locations were divided into three categories: on the HS’s side, on the LS’s side, and between the two. Statistics on the proportion of these three categories help us find the relationship between task set location and LS performance. The proportion of HS-side task set location is shown as a line chart.

The Fig. 8.5 shows that the task unit was set on the HS side in Group 2 far less often than in Groups 1, 3, and 4. The task set in Group 2 was between the two students almost 85% of the time, allowing the LS to obtain task information effectively and participate in CPS. The HS did not control the task set location. At the beginning of the task, the HS chose to put the task set between the peers and made reasonable use of the blank paper given by the researchers.

Fig. 8.5

The proportion of task set on the side of HS

In the other three Category As, task placement was generally on the HS’s side, and LS did not participate actively, leading to the following reasonable conclusions. First, task set location impacts peers’ interaction enthusiasm; when LS had more access to task information, they could participate in the discussion more actively. Second, when the academic performance gap between the peers was large, HS were likely to take absolute control of task set location, viewing the task set as their own and ignoring pair collaboration. Task set position directly reflected HS’s attitude toward pair collaboration.

In Group 5, the side task set appeared less often on the HS side as the academic performance gap between HS and LS narrowed, appearing more often on the LS side or between HS and LS. This study found that in almost all groups, either HS had great decision-making power over the task set location or LS had little desire to control the task set. As the performance gap narrowed, the HS respected the LS significantly more and became more concerned about the LS’s task understanding.

Task set location was mainly controlled by the HS. When the task set was always on the HS’s side, the LS could not gain task initiative and behaved correspondingly. As the performance gap between the peers narrowed, the LS followed one of three paths: never winning the task list and being controlled by the HS; fighting for the task set and then giving up; or striving for the task set and ultimately succeeding.

In the first three groups’ CPS processes, the LS did not show dissatisfaction with the task set location. The HS decided the task set position and the LS neither tried to change the task unit nor actively proposed writing answers on the task set. In Group 2, the task set was not placed on the LS’s desk but between the two on the HS’s initiative. HS generally controlled the task set; when the academic performance gap between the two was very large, LS could not control the task set (Figs. 8.6 and 8.7).

Fig. 8.6

The picture of Group 1

Fig. 8.7

The picture of Group 3. Note The task set is on the desktop for HS

The episode above depicts the task set in Group 4. The girl was the HS, while the boy was the LS. After two minutes of discussion, the boy snatched the task set but it was immediately taken back by the girl.

LS: my God! This is your painting! Not even legs! I’ll give the painting some more (the LS tries to take the task set)

HS: don’t move! This is my own creation!

LS: what’s the creation? (the LS take the task set successfully)

HS: what are you doing? It cannot be destroyed!

LS: This is the father of the seventh grader, so I’ll draw all of them for you. Damn it, no one can get the task set. (The HS tries to get the task set back)

HS: let’s all add feet to the painting. (The HS gets the task set successfully)

For about 33% of the time, the LS fought for the task set, adopting a tough attitude and saying, “Either we are half [or] alone.” The HS compromised and guided the LS about what he should write (“Here you write, one is brother, one is sister”). However, the boy had few task-solving views and ideas and asked the girl, “a brother, and then what?.” After the girl answered, the boy said nothing that promoted task development. After 30 s, the girl took back the task set and held it until the end of the task (Fig. 8.8).

Fig. 8.8

The picture of Group 4

Group 5’s task set was almost always on the HS’s side, but the two peers’ interaction was slightly more positive than in the previous four groups and their dialogue lasted longer, accounting for about 80% of their time. During the discussion, the LS actively looked at the task to get task information (Fig. 8.9).

Fig. 8.9

The picture of Group 5

In Group 6, the task set was on the HS’s desk for the first six minutes. Later, the LS began to fight for the task set and succeeded in getting it. When the HS proposed modifying the answer, he was opposed by the LS and did not resist. The LS held the task list for the remaining time. Their interaction was positive, with both peers voicing their views and refuting those with which they disagreed (Fig. 8.10).

Fig. 8.10

The picture of Group 6. Note In this picture, the front student is the LS, and the back student is the HS

The HS no longer insisted on holding the task set in the last three groups. In groups 8 and 9, the HS took the initiative to put the task set on the LS desktop or between the two. In Group 8, the LS expressed that he wanted to write the answer on the set and strove for the initiative to complete the task. Although the LS failed, their behaviour did not appear in the first seven groups. In Group 9, the HS showed great respect for the LS, actively suggesting that the LS write the task answer.

The two peers’ performances were significantly different in Category A and Bs, indicating that when the academic performance gap between peers was controlled within a reasonable range, HS paid more attention to the LS’s views, and both had equal opportunities to get the task set, giving LS a stronger sense of participation in the CPS process.

8.4 Discussion

8.4.1 Discussion

• 1. Students in heterogeneous pairs perform differently in peer interactions in collaborative mathematics problem solving.

HS and LS had different mathematics problem solving performances in different pairing modes, mainly manifesting in two aspects: the amount of utterance between them and their interaction performance indicators. HS had a relatively large amount of utterances than LS in the task completion process and more frequently used the interaction indicators “order,” “negation,” and “evaluation.” LS’s had fewer utterances than HS and more frequently used “repetition” and “proposal” interaction indicators. In most cases, the HS controlled the discussion throughout the interaction; the LS, having neither the opportunity nor the courage to express their views, gradually deviated from the group discussion and became silent, consistent with Hou’s (2017) findings.

Some scholars divide help-seeking into two types: execution requests and tool requests. Execution requests are oriented by dependence, while tool requests are oriented by mastery. The students with strong mathematical ability tend to offer instrumental help, while those with low mathematical ability offer executive help. This conclusion explains the differences in dialogue indicators between HS and LS. Nelson Legal and colleagues suggest that low-level students may not be effective helpers because they may not only lack the skills to recognise their need for help, but may also be unable to seek the best learning help.

• 2. Based on this study's collaborative mathematics problem solving exercise, the interaction in heterogeneous peer collaboration groups is affected by the academic performance gap between peers.

In collaboration, intellectual authority is more stable while social authority is more dynamic (Langer-Osuna et al., 2020). In the peer collaboration groups, the difference in academic level between HS and LS affected the group members’ interaction performance. The nine peer groups were divided into two categories based on interaction time. Category A, which included Groups 1–4, had the least interaction time; Category B (Groups 5–9) had the most. There were differences in interaction performance between Categories A and B.

• (1) Peer collaboration groups with a large academic performance gap rarely interacted.

In Category A groups, peer interaction was less, and LS seldom participated. The LS used significantly more “repetition,” “proposal,” and “off-task” interaction indicators than HS, who accounted for a higher proportion of “order” and “negation” indicators. In Category A groups, the task set was most often located on the HS’s side, and sometimes the LS could not see the task information. In Category A groups, LS were students with learning difficulties.

According to the existing research, in collaboration groups, students with learning disabilities hold a negative attitude towards other group members in terms of willpower and behaviour characteristics, which seriously affects their communication in the group (Li, 2015). Other studies have also pointed out that the communication level between students with learning difficulties and their peers is relatively low (An et al., 2017). In addition, in terms of their psychological characteristics, students with learning difficulties mainly manifest hostility, interpersonal sensitivity, and so on. Low-level students have difficulty actively integrating themselves into collaborative mathematical problem-solving discussions without their peers’ support and help, which hinders the formation of an active peer group. Case interviews in other studies revealed that many HS are unwilling to communicate with LS. They lack the psychological preparedness and skills to communicate with LS, and they worry that LS will negatively impact their learning and task completion (Jiang, 2007).

If the performance gap between peers is too large when collaborating to solve mathematical problems, the HS will put forward more views and opinions. LS have difficulty understanding and accepting new concepts and cannot keep up with the progress, so they passively or actively give up the opportunity to participate in the discussion (Zeng & Zhang, 2016), reducing peer interaction and collaboration quality.

• (2) When the academic performance gap is reasonable, group interaction performance is better.

In Category B groups, the LS’s academic performance was slightly higher than in Category A groups, and the academic performance gap between the peers was relatively small, leading to the following observations. First, the number of utterances between the peers increased significantly, and the interaction was more positive. Second, from a dialogue indicator perspective, HS used far fewer “order” and “negation” indicators; LS had more space to express their opinions, significantly improved their views and explanations, used more “order” indicators, and could even lead the interaction. Third, from a task set location perspective, the task set no longer appeared only on the HS’s desktop but spent more time between peers or on the LS’s desktop.

Some scholars divide students into excellent, middle, and poor categories and set control and experimental groups to explore the effects of collaborative learning on the three types. Research has found that excellent students have the highest acceptance level in peer relationships. Although there is no significant difference between the control and experimental groups, students with learning difficulties had lower acceptance levels than the control group in the collaborative learning process, while middle students’ acceptance level was higher (Jiang & Tan, 2011), which can explain the difference in Category B and A students’ interaction performance to a certain extent.

• 3. In pair CPS in mathematics, task set location affects peer collaboration.

In Category A, Group 2 showed relatively abnormal performance, with high dialogue time and high LS participation. The video revealed that the task set was placed between the peers in Group 2, and the LS had more opportunities to understand and think about the task. This study also found that in Category B groups, the task set was less often placed on the HS’s desktop than the LS’s, and the interaction between them was more positive. LS gradually strove to control the task set. As mentioned above, LS who could not see the task information found it difficult to participate in the CPS. When the peers were in a “no communication” state, the LS could not get the task information and had even more difficulty concentrating on it. The widely-used Team Games Tournament collaborative learning method points out that every member needs to know the task materials to contribute to higher team scores. Therefore, task set location is extremely important and must ensure that both students can see and understand the task information to collaborate better.

8.5 Conclusions and Implications

This study has mainly focused on student interaction in pair collaboration. It found that LS did not find it easy to participate in cooperative discussions and that the discussion between the peers was sometimes unequal. The following implications were obtained.

• 1. Cultivate students’ sense of collaboration.

A good collaborative group can improve students’ performance because group members can encourage and help each other in peer learning (Slavin, 1991). HS can dominate the collaboration process, and their attitudes and expressions affect, to some extent, how well LS can participate. Therefore, before collaboration, students should be trained to realise that collaborative problem solving requires every student’s participation and is not a one-person show.

Some studies point out that collaborative groups should establish a positive goal of interdependence through mutual learning goals, including learning materials, to ensure all group members can learn and understand the specified materials (Johnson & Johnson, 1999). Students need a positive sense of collaboration to ensure fairness in the collaborative problem solving process, so everyone can learn and understand the task materials.

If LS raise questions or suggestions in the peer collaboration process, HS should give timely feedback and reflection. Webb (1991) pointed out that how students’ requests for help are responded to is more important than the kind of help they get and that accepted help is only effective when applied to solve problems.

• 2. Teachers should make appropriate interventions.

In the “no intervention” state, some groups cannot cooperate well and lack collaboration consciousness, requiring teachers to make timely adjustments. Students with learning disabilities’ low-level information dependence makes it difficult for them to participate actively, and external intervention is needed (Chen, 2020; Zhang et al., 2021). Johnson pointed out that teachers should supervise students’ learning and intervene appropriately to develop students’ interpersonal communication and group collaboration skills (Johnson & Johnson, 1999). Students do not interact spontaneously in classroom groups (Fuchs et al., 1994) and must be trained before collaboration or given proper teacher guidance during it to improve the enthusiasm and quality of their discussion.

Besides intervening in organisational collaboration, teachers should also consider students’ evaluation methods. In peer groups with a large gap in academic performance, one of the reasons for LS’s low collaborative behaviour efficiency is that they are not recognised; a single evaluation based on academic performance will aggravate this phenomenon. Evaluations should be diverse and consider students’ behaviour and performance from many aspects and angles, not only academic performance.

• 3. A collaboration group’s organisation shall meet the reasonable matching mode.

Task type, group composition, and teacher support are the main reasons informing the effectiveness or ineffectiveness of collaborative learning (Kahilainen et al., 2007). Most studies advocate intra-group heterogeneity. Vygotsky’s zone of proximal development theory advocates that students collaborating with more capable students benefits both. However, in pair collaboration to solve mathematical problems, if the academic performance gap between the peers is too large, the communication between them will be unequal, the LS’s expression will be unrecognised or interrupted, and the LS’s thoughts cannot keep up with HS’s, making it difficult to continue the interaction and achieve the benefits of collaboration. When the academic performance gap between peers narrows, their interaction is relatively positive, and their utterances can collide, enabling them to exchange views and agree on goals through mutual debate and compromise and allowing all students, especially those with low knowledge levels, to benefit from collaboration.