Keywords

10.1 Introduction

Since the beginning of the twenty-first century, international organisations such as the United Nations Educational, Scientific, and Cultural Organization (UNESCO) and the European Union and countries such as the United States and Australia have put forward twenty-first-century competency frameworks to develop the knowledge, abilities, and attitudes expected of citizens, including their capacity for collaboration and problem solving (Peng & Deng, 2017). Similarly, China’s Mathematics Curriculum Standards for Compulsory Education (Ministry of Education of the People’s Republic of China, 2012) emphasises developing students’ collaborative communication and problem solving capacities, indicating it is beginning to attach importance to collaborative problem solving (CPS) in mathematics and advocate problem solving orientation and problem-based and collaborative learning (Yu & Cao, 2017). Many large international assessment programmes such as PISA and ATCS21S are beginning to focus on measuring students’ collaborative problem solving capacity.

Both PISA 2015 and ATCS21S deconstruct collaborative problem solving capacity as collaboration (social skills) and problem solving (cognitive skills); from these two concepts, the origins of collaborative problem solving are long-standing. The effectiveness of collaborative/cooperative learning has been of interest to academics since the concept’s inception. Through a review of hundreds of studies, Bossert (1988) found that students in collaborative learning classrooms performed academically at least as well as and often better than those in traditional classrooms. In recent years, some meta-analyses have found that students achieve higher grades by working in small groups than in individual learning—e.g., in computer-supported collaborative learning (Chen et al., 2018) or face-to-face collaborative learning (Kyndt et al., 2013). It is widely agreed that collaborative activities can facilitate student learning, and the collaborative group learning model is becoming common in classrooms worldwide.

The mathematics classroom plays an irreplaceable role as a key site for developing students’ collaborative problem solving skills. In mathematics classroom, there was a transfer from individual problem solving to collaborative problem solving (Pruner & Liljedahl, 2021). Collaborative problem solving capacity cannot be developed through a specialised subject and is not a separate discipline taught on a case-by-case basis through specific teaching content (Bai & Lin, 2016). Xu et al. (2020) suggested that subject-based and interdisciplinary curricula are important basics for developing collaboration literacy. Furthermore, due to the nature of the mathematics curriculum, CPS is required in mathematics classrooms, and mathematics tasks can be provided to foster it (Kong & Zhao, 2017). Collaborative mathematics problem solving can facilitate the development of related competencies.

CPS in mathematics could develop students’ relevant skills, such as collaboration, problem solving, etc. It has been well noted that teachers play critical roles in such processes. Teachers act as designers and organisers of collaborative activities and are, for students, academic experts and managers in mathematics classrooms (Davidson, 1990). However, many researchers argue that simply grouping students does not guarantee that collaborative learning or collaborative problem solving will occur. More effort is needed from teachers to enhance further students’ collaborative and problem solving capacity and provide them with adequate learning opportunities (Chen & Zhang, 2014; Hou, 2017). In group work, teachers also need to act as facilitators and guide students’ collaboration through diagnosing and intervening, involving each student in the group’s task to enhance the quality of their discussions and further the teaching objectives.

Collaborative learning was introduced later in Chinese classrooms, around the beginning of this century, and experienced difficulties initially. Group learning was not developed sufficiently (Wang & Wu, 2012), and teacher-group interactions were sorely lacking (Cao & He, 2009). In a 2013 survey of teachers from 13 provinces in China, 48.9% said they often carried out “collaborative learning in groups under the guidance of teachers” and 60.3% said they often organised discussion activities for students (Shi & Wang, 2018). With the popularisation of collaboration, the proportion of teacher–group interactions in China’s mathematics classrooms has increased significantly in recent years; however, some studies have found that teacher intervention needs to be improved. A lack of interactions between teachers and individual students or groups has been noted (Yu & Cao, 2018), and guidance and evaluation of group communication are relatively lacking (Dong et al., 2013). In addition, there have been few empirical studies on collaborative mathematics learning in China (Wu et al., 2020), and there is still a lack of research on teacher intervention in collaborative learning, especially on the effect of teacher intervention on students’ classroom performance.

Open-ended mathematics tasks are chosen in this study to investigate the impact of teacher intervention in collaborative problem solving in mathematics. Research has found that open-ended problems are suitable for collaborative learning (Wang, 2016), facilitate students’ reasoning arguments and mathematical communication (Kosyvas, 2016), and help develop students’ mathematical thinking and interest in learning.

Teaching and learning enhancement should be promoted to foster students’ collaborative problem solving skills. This chapter analyses the current state of the situation and problems in a targeted manner based on an understanding of the current situation of teacher intervention in collaborative problem solving classrooms in mathematics. The study digs deeper into the situation’s causes and is conducive to promoting collaborative problem solving activities in a scientific and rational direction. This chapter investigates how teacher interventions affect students’ collaborative problem solving through experiments. It then selects typical cases to discover Chinese mathematics teachers’ advantages and weaknesses in collaborative problem solving, mainly using coding frameworks such as teacher intervention focus and means.

10.2 Literature Review

10.2.1 Definition of Teacher Intervention in CPS

Webb (2009) classified the teacher’s role in collaborative problem solving into several dimensions: preparing students for collaborative work, forming groups, structuring the group work task, and influencing student interaction through teachers’ discourse with small groups and the class.

Previous research has shown that the effectiveness of collaborative learning depends largely on the quality of student interaction. Based on a literature review, Kaendler et al. (2015) compiled a framework for Implementing Collaborative Learning in the Classroom (ICLC). The framework clearly describes the teacher’s role in facilitating student collaboration, dividing teacher–student interaction into two dimensions: the student and teacher levels. The student collaboration level includes three phases—pre-active, inter-active, and post-active—while the teacher level describes five competencies across all implementation phases. The framework in Fig. 10.1 argues that the various teacher competencies are based on teachers’ professional knowledge and teacher beliefs.

Fig. 10.1
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The ICLC framework by Kaendler et al. (2015)

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In earlier studies, the definition of teacher interventions did not go beyond teacher-initiated activities. However, more recent researchers have included student-initiated interventions (e.g., Kajamaa et al., 2020).

Our study operationally defines teacher intervention in collaborative problem solving as a verbal intervention at the individual or pair (group) level initiated by the students or teacher, excluding non-task-related instructions, during pair and group collaboration. The interventions defined here only focus on the interactive phase of the collaboration, excluding the pre-active and post-active phases. We also leave out discourse, such as reading out tasks to the whole class, explaining task requirements, or using directive language.

10.2.2 Review and Analysis of Empirical Research on Teacher–Group Interaction in China’s Mathematics Classrooms

The concept of collaborative learning originated in the Western world but has been applied in China for many years. In the 1980s, scholars in China began to translate Western experimental research on collaborative learning; by the late 1980s and early 1990s, research and experiments on collaborative learning had emerged in China (Zeng & Tian, 2014).

In 2001, the State Council of the People’s Republic of China promulgated the Decision of the State Council on the Reform and Development of Basic Education, which specified that collaborative learning should be encouraged to promote students’ mutual communication and development (State Council of the People’s Republic of China, 2001) Since then, collaborative learning has been fully promoted in mathematics classrooms and has been the subject of more studies. Collaborative learning has become a top topic in China’s mathematics curriculum reform, with China’s Compulsory Education Mathematics Curriculum Standards (2011 edition) intentionally infiltrating collaborative problem solving skills (Kong & Zhao, 2017). In recent years, the spread of new educational philosophies and technologies has informed new classroom teaching models—such as problem-based learning, problem solving teaching, and flipped classrooms—that emphasise group collaboration and require collaborative learning and problem solving activities.

Although group work is in full swing in Chinese mathematics classrooms, research into teacher–group interactions started late, and there are generally fewer teacher–group interactions than in other countries. The earliest study in China on teacher–group interaction in mathematics classrooms (Cao & He, 2009) coded videotaped classroom interactions into four categories (teacher–individual, teacher–group, teacher–class, and cross-interaction) and found that teacher–group interaction was relatively lacking. Subsequent research relying on that coding scheme studied teacher–student interactions in mathematics classrooms, yielding the findings in Table 10.1.

Table 10.1 Related research on teacher–group interaction in China
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As seen from Table 10.1, the percentage of teacher–group interaction for the whole class was lower in earlier years. Researchers identified a lack of teacher–group interaction at that time, noting that collaborative learning in groups was less developed in mathematics classrooms and teachers had fewer interactions with students in group work. While the students worked together, teachers prepared content for subsequent lessons or patrolled the classroom without any interactions. Teachers only interacted with students when they found something wrong (Cao & He, 2009). In some classrooms, teachers would mainly ‘observe,’ ‘nod,’ and ‘point’ during their rounds, spending very little time with each student group (Wang & Wu, 2012).

The table also shows that as the curriculum began to call for more group work, teachers gradually took on more group guiding work, resulting in a significantly higher percentage of teacher-group interactions.

10.2.3 Teacher Intervention During Collaboration

Earlier studies have researched the effect of teacher intervention during students’ collaboration, but their results are inconsistent. Most found a positive correlation between teacher presence in intervention and high levels of student knowledge processing, while others reported neutral findings. For example, Hogan et al. (1999) found that while groups performed moderately well overall in scientific reasoning when the teacher intervened, some presented higher reasoning when there was no intervention, and others presented lower levels. However, a few studies have also found a negative correlation between teacher presence in intervention and high levels of student knowledge processing. The timing and type of teacher support are possible explanations for the negative correlation results; Ros (1993) found that longer teacher–student contact correlated with lower levels of student knowledge processing, making brief acts of teacher–student interaction appear more desirable. Further, some studies have found that teacher presence and absence in the intervention are unrelated to student task engagement, while others have found that both are positively related to student task engagement. For example, Van de Pol et al. (2015) found that students could sustain task discussions well without the teacher’s presence if the teacher had adequately and appropriately designed the instructional activity.

The equality of teacher intervention has been considered. Van Leeuwen et al. (2013) found that teachers get more involved with high-activity groups, while Pietarinen et al. (2021) found no correlation between prior competence and teachers’ guidance, with teachers preferring to guide groups they perceived as motivated and willing to collaborate.

While the development and application of ICT (Information and Communications Technology) have resulted in many proven effective teacher intervention tools for computer-supported collaborative learning (CSCL) (e.g. Schwarz et al., 2021), in China, there is still a long way to go.

10.3 Current Study

10.3.1 Research Questions

This chapter studies teacher intervention in students’ collaborative mathematics problem solving. Although similar studies have been conducted in several countries and regions, there is a lack of current research on teacher intervention at the elementary education level in China. While computer-supported collaborative learning (CSCL) has received more attention recently, this research studies teacher intervention in face-to-face collaborative problem solving, with open-ended tasks chosen to achieve better collaborative efficiency. While most studies have found that teacher intervention greatly impacts collaborative problem solving, a few have found that students can do as well or even better when there is no intervention; thus, intervention effectiveness should and has yet to be verified.

Previous research on collaborative learning and collaborative problem solving has mainly studied interventions at the behavioural level rather than investigating how teachers’ interventions occur.

Based on the preceding discussion, the research questions for this paper are as follows:

RQ1: What is the effectiveness of teacher interventions on the outcomes of students’ pair and group collaborative mathematics problem solving?

RQ2: What are the similarities and differences in the characteristics of teacher interventions with different effects on students’ pair and group collaborative mathematics problem solving?

10.3.2 Research Design

This study adopted a mixed research approach with a Sequential Explanatory Design that combined quantitative and qualitative research methods. Quantitative tools were used to discover phenomena (RQ1), and qualitative research methods were used to explain and further explore those phenomena (RQ2).

A quantitative study was conducted to investigate the effectiveness of teacher interventions in collaborative problem solving by comparing the pair and group work scores of students in the intervention and control groups and analysing which aspects of the interventions were effective based on the response scores. Based on the pre-test scores, the pairs or groups were divided into different structures (high-score homogeneous group, low-score homogeneous group, and heterogeneous group). The effectiveness of teacher intervention was determined by comparing the performance of the pairs and groups within the different structures.

Based on the quantitative results, a typical teacher was selected to conduct a case study of classroom teaching in the intervention group, using an Interpretational Analysis data analysis model. Based on Van Leeuwen et al.’s (2013) framework, the teacher interventions were coded to explain how they occurred and affected students.

The case study needed to ensure the integrity of the context. This chapter relies on the original video recordings to analyse the data. It focuses on tacit knowledge, describing non-verbal clues as well as possible to recreate the real classroom context. The coding section draws on previous research on teacher intervention focus categories, intervention means, intervention initiators, and intervention targets to compare intervention situations and effects across teachers. In addition to analysing teacher–student interaction behaviours in the classroom, the analysis and discussion section also analyses parts of the teacher interviews, using different data sources for crystallisation.

10.3.3 Participants

A purposive sampling method is applied in the current research, all participating teachers are interested and have experience in collaborative learning, which grants the investigation of teachers’ intervention is aligned with their daily classroom. The participants were 292 Grade 7 students selected from eight classes taught by four teachers in two secondary schools in District T of City B. The differences in mathematics performance between the two classes taught by the same teacher were not statistically significant, as shown in the following Tables 10.2 and 10.3. Each teacher designated one of their two classes as the control class and the other as the intervention class. They then divided each class into small groups based on mathematics performance to ensure that the average scores of each pair and group were not significantly different; adjustments were made based on discipline. Students first worked in pairs, and then two (or three) pairs worked in one group. Due to space limitations and the need to involve all students, several pairs of three students and groups of five to six students emerged in each class.

Table 10.2 Basic information for each class
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Table 10.3 Mathematics performance of each class
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Mathematics performance was based on the current semester’s mid-term exam results; however, some students had missing grades, which were set as missing values. A t-test found no statistically significant differences in Table 10.5, despite some differences in the mean grades of each teacher’s control and intervention groups.

10.3.4 Data Analysis

The data selected for this chapter consist of classroom videos, task sheets, and video interviews with teachers. Data on pair and group collaborative problem solving were selected (Task 1 and Task 2, respectively, see Appendix for details). The data analysis consisted of scoring students’ collaborative problem solving outcomes and coding teacher interventions, including their intervention focus, means, initiators, and targets. In the problem solving process, students in the same pairs and groups reported their collaborative problem solving outcomes on the same task sheet, which served as the basis for scoring their collaborative problem solving results. The scoring framework was piloted and polished to ensure it was clear and covered the performance of all students, with items categorised to facilitate exploring specific student performances. The data for analysing teacher intervention characteristics were primarily derived from the follow-up teacher video, using the groups’ videos for further confirmation. The frameworks for teacher intervention of focus, means, initiators, and targets were drawn from previous research.

10.3.4.1 Scoring of Task Sheet

The scoring schemes were based on pair (group) collaboration tasks; the score for each item represents whether the student understood the relevant task content reasonably well and represented it clearly. The task sheet scoring scheme is shown in Table 10.4.

Table 10.4 Pair task sheet scoring scheme
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Table 10.5 Group task sheet scoring scheme
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The marking scheme for the paired task was based on Task 2, where the ‘Comprehension of task’ items corresponded to questions about ‘one of the five people is a Year 7 student’ and ‘five people living in a house.’ The ‘Mathematical performance’ items tested whether the ages of the other four people were calculated correctly. The ‘Character roles’ and ‘Character relationships’ items tested whether students reasonably represented how the five people in the house were related, with ‘Character relationships’ concerning the generational age difference. ‘Expression’ items corresponded with the instruction, ‘Write a paragraph explaining your answer,’ with each item’s scoring depicting how the pairs performed at each level. The scores for each item are added together to give an overall score for pair collaboration problem solving, ranging from 0 to 5, discretely (Table 10.5).

The group task sheet scoring scheme was designed based on Task 1, where the item ‘Number of rooms’ was based on the task’s requirement that an ‘apartment has five rooms,’ ‘Marking of functions’ and ‘Geometric drawing’ corresponded to ‘Label each room and show the dimensions (length and width) of all rooms,’ ‘Geometric drawing’ tested students’ drawing performance, and ‘Area’ corresponded to the ‘total area is 60 m2’ item. As with the pair tasks, the total scores for the group tasks were discrete (0–5 points).

10.3.4.2 Analysis of Teacher–Student Dialogue

Units of Analysis

The dialogue units of analysis were closely related to the study’s research questions. This chapter focuses on verbal teacher–student interactions, where the teacher moved between groups within the class, and who initiated the dialogue (i.e., uttered the first sentence in the conversation); thus, the units of analysis were divided based on each conversation. Each sentence spoken by the teacher may have a different intervention purpose, form, and target. The units of analysis were divided based on those in a typical two-person face-to-face discourse; teacher-student conversations always followed rounds, so the units of analysis were divided mainly by speaker changes. It is important to note that a turn sometimes contained more than one discourse and that an utterance was divided by ‘perceptible pauses’ (commas or full stops) in the transcribed text (van Boxtel et al., 2000).

Dialogue Initiators and Intervention Targets

The dialogue initiators classification was mainly based on Chiu’s (2004) teacher intervention initiators classification, where the unit of analysis for a dialogue is the first person to speak in a single teacher–student dialogue (usually with multiple talking rounds). The categories were student-initiated and teacher-initiated.

In this study, teachers were suggested not to conduct class-oriented interventions or guidance. Therefore, the intervention targets were categorised depending on whom the teacher was talking to—individuals, pairs, or groups. Every turn was coded as a unique intervention target.

10.4 Results

10.4.1 Results of Pair Collaborative Problem Solving

10.4.1.1 Overall Results

First, based on the design, the results of pair tasks were compared between the intervention and control classes. Figure 10.2 shows that the intervention classes’ scores were higher than the control classes,’ indicating the intervention class students generally performed better in pair collaborative problem solving.

Fig. 10.2
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Comparison of pair collaborative problem solving results

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Further statistical tests compared the means for the intervention and control classes. Since neither data group followed a normal distribution, a non-parametric test (Mann–Whitney U test) was used to test the differences between the two groups.

Combining Cohen’s d and p-values, Table 10.6 shows that the intervention classes outperformed the control classes in all items except ‘Character role,’ for which there was no difference. The most significant differences were found in ‘Mathematical performance,’ which showed a moderate difference based on effect size, and ‘Expression,’ which showed a moderate difference with borderline significance. The differences in the other items were not statistically significant and showed lower validity. Overall, the differences in the total scores of the two groups were marginally significant with moderate validity.

Table 10.6 Scoring result of pair collaborative problem solving
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These results show that the teacher intervention had its most significant effect on ‘Mathematical representation,’ as evidenced by the fact that more pairs calculated correctly and expressed their responses clearly on the task sheet. Overall, teacher intervention improved student pairs’ response outcomes in collaborative problem solving tasks.

10.4.1.2 Comparisons of Pairs

The pairs were divided into four pair types based on pre-test scores and pair structure: high-score homogeneous, middle-score homogeneous, low-score homogeneous, and heterogeneous (Table 10.7).

Table 10.7 Basic information of groups of different pair structure
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Based on Levene’s test, the was a variance in homogeneity (p = 0.196); ANOVA could be performed to compare the differences in mean scores of different pair structures.

Table 10.8 shows a significant difference in mean values between the different pair structures. A further one-by-one comparison of data from the different groups was performed using the Least Significant Difference (LSD) method, the results of which are shown in Table 10.9.

Table 10.8 Result of ANOVA
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Table 10.9 One-to-one comparison of the different structures of pairs by LSD
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Table 10.11 shows significant differences between the low-score homogeneous pair and the medium- or high-score homogeneous pairs and marginally significant differences between the heterogeneous pair scores and the low-score homogeneous and high-score homogeneous pair scores. All other differences were not significant.

A 4 (pair structure) × 2 (intervention or not) multifactorial between-group experimental design further explored the intervention’s effects on pairs with different pair structures.

Correcting for the model term (F = 2.576, p = 0.016), the model was significant, where the experimental term (F = 4.142, p = 0.044) and the pair structure term (F = 2.598, p = 0.055) were statistically significant, while the crossover term was not. The above between-subjects effect test indicated that the model was reasonable and explained 11.7% of the variance (\({R}^{2}=0.117\)) (Table 10.10).

Table 10.10 Between-subjects effect test
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Table 10.11 Mean and standard deviation performance of pairs of different levels
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Based on Fig. 10.3, the performance of the middle- and high-score homogeneous pairs was higher in the intervention classes than in the control classes. The performance of the heterogeneous pairs in the intervention classes was somewhat higher than in the control classes. Nevertheless, the performance of the low-scoring homogeneous groups in the intervention classes lagged behind their control classes peers. The intervention most improved the middle- and high-score homogeneous pairs’ performance, particularly for the heterogeneous group, while the low-scoring homogeneous group’s performance showed a decreasing trend.

Fig. 10.3
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Score of different pair structures

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10.4.1.3 Comparisons of Differences in Interventions Across Teachers

Comparing the pair collaborative problem solving results for the intervention and control classes revealed that students in the intervention class outperformed those in the control class. Regarding each teacher’s intervention effects, Teachers A, C, and D’s intervention classes outperformed their control classes, while Teacher B’s classes did the opposite (Table 10.11).

A comparison of means revealed that Teachers A, C, and D’s intervention group class pairs scored higher on the collaborative task and had less in-class score differentiation, with Teacher D’s class showing the most significant improvement. In contrast, Teacher B’s intervention class pairs scored lower on the collaborative task and had a more pronounced divergence.

10.4.2 Results of Group Collaborative Problem Solving

10.4.2.1 Overall Results

The overall evaluation of the collaborative problem solving results in groups is shown in Fig. 10.4, based on the evaluation framework mentioned earlier.

Fig. 10.4
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Comparison of group collaborative problem solving results

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The figure shows that the intervention classes outperformed the control classes in group collaborative problem solving. As neither group of classes’ performance was normally distributed, a non-parametric test was used to test the difference between the two data groups.

Based on Table 10.12, the intervention class outperformed the control class on all items except ‘Marking of functions.’ Combining Cohen’s d and p-values, the most significant differences were found for the ‘Number of rooms’ item, which showed a high level of validity, and the ‘Geometric drawing’ item, which was borderline significant and showed a medium level of validity. The differences in the other items were not statistically significant and showed lower validity. Generally, comparing the two groups’ total scores revealed significant differences with moderate validity.

Table 10.12 Scoring result of group collaborative problem solving
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The above results show that, with teacher intervention, more students correctly marked the number of rooms, indicating they understood the concept of ‘room’ in the task through their experience and could solve questions such as, ‘Is a balcony an apartment room?’ This will be discussed in more depth later.

In addition, some groups with more than four members appeared to lag behind the four-member groups’ overall performance but outperformed the intervention classroom group with more than four members and the control class, as shown in Table 10.13.

Table 10.13 Scoring result of groups of different sizes
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10.4.2.2 Comparisons of Groups

Based on the pre-test scores, the student groups were divided into three group structures: high-score homogeneous, low-score homogeneous, and heterogeneous. A 4 (group structure) × 2 (intervention or not) multifactorial between-group experimental design was used to explore the intervention’s effects on pairs with different pair structures further (Table 10.14).

Table 10.14 Basic information of groups with different group structures
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Adjusting the model term (F = 2.069, p = 0.081) showed the model was borderline significant, where the experimental group term (F = 4.442, p = 0.040) was statistically significant, while the group structure term and the crossover term were not. The above between-subjects effect test indicated that the model was reasonable and explained 14.3% of the variance (R2 = 0.143) (Table 10.15).

Table 10.15 Between-subjects effect test
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Figure 10.5 shows that the intervention classes’ low-score and high-score homogeneous groups outperformed the corresponding two control class groups, with no significant difference in the groups’ performance. In other words, the intervention improved the performance of the low- and high-scoring homogeneous peer groups most significantly, while there was nearly no change in the performance of the heterogeneous group.

Fig. 10.5
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Score of different group structures

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10.4.2.3 Comparisons of Differences in Interventions Across Teachers

As can be seen from the above results, Teachers A, B, and D’s intervention classes performed significantly better than their control classes, while Teacher C’s intervention and control classes had no significant performance difference. Group performances in intervention classes were less discrete than those in the control classes in all teachers’ classes, except for the two classes taught by Teacher B (Table 10.16).

Table 10.16 Mean and standard deviation performance of pairs of different levels
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10.4.3 Results of the Case Study

As there were no statistically significant differences in academic performance between classes taught by the same teacher, the effect of teacher intervention on CPS outcomes can be determined by comparing the performance of the intervention and control classes. The previous statistical analysis revealed that teacher interventions generally contributed to students’ collaborative pair and collaborative group problem solving outcomes. However, there were exceptions. The intervention class taught by Teacher B had lower overall pair CPS performance than the control class, while there was little difference in Teacher C’s two classes’ collaborative group problem solving.

The above discussions were oriented toward the peer or group collaboration results, and the intervention’s impact and effectiveness were judged solely by the results. Cases had to be selected and analysed in depth to explore teacher interventions’ impacts further. Accordingly, two teachers, Teachers B and D, were selected. Based on student response results alone, Teacher B’s intervention was the least effective, with the intervention class performing poorer than the control class, while Teacher D’s intervention was the most effective. The basic information about the two teachers’ effective verbal interventions is shown in Table 10.17.

Table 10.17 Number of teacher intervention
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In general, the teachers initiated most interventions in both classes (Fig. 10.6). However, the percentage of intervention initiators in the two teachers’ classes differed significantly at different stages. In Teacher B’s class, the students initiated most of the pair collaboration stage conversations, while the teachers initiated the vast majority of group collaboration stage conversations. In Teacher D’s class, the students initiated nearly two-thirds of the pair collaboration stage conversations, while the teachers initiated half of the conversations in the group collaboration stage. In Teacher D’s class, nearly two-thirds of the pair collaboration phase conversations were teacher-initiated, while teachers and students each initiated half of the group collaboration phase conversations.

Fig. 10.6
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Initiator of dialogue

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In the two collaborative problem solving phases, the teachers’ intervention foci were slightly different, with the interventions Teacher B initiated focusing on cognitive activities (17/20 in the pair phase and 12/23 in the group phase) and those Teacher D initiated focusing on social activities (19/34 in the pair phase and 17/26 in the group phase).

In the different problem solving stages, all student-initiated questions concerned cognitive activities. Most of both teachers’ subsequent interventions were also directed toward cognitive activities, especially for Teacher B, who directed almost all interventions at cognitive activities (Pair stage Teacher B: 17/18, Teacher D: 9/13; Group stage Teacher B: 4/4, Teacher D: 12/17) (Fig. 10.7).

Fig. 10.7
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Target of teacher intervention

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Both teachers generally prefer to intervene in the whole pair/group. Teacher B’s pair stage interventions focused more on individuals, while her group stage interventions focused more on the whole group (Table 10.18). Teacher D’s interventions were almost the same at both stages, mostly focusing on pairs/groups while still paying sufficient attention to individual students (Fig. 10.8).

Table 10.18 Performance of each pairs (average in groups) and groups
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Fig. 10.8
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Distribution of teacher intervention

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Figure 10.7 shows the disruption of intervention by Teachers B and D, indicating that both teachers initiate interventions among nearly equally groups. As the teachers claimed that they made students in different groups have similar performance in previous tests, the current study could not support information on the correlation between the previous test score and teachers’ favour in intervention. Compared with students’ CPS performance, there was possibly a negative correlation between distribution and performance. It could be that both teachers preferred to pay more attention to students in difficulty, which seemed ineffective for difficult pairs and groups.

10.5 Discussion

10.5.1 Intervention Was Generally Effective, but Limited for Heterogeneous Groups

Regarding collaboration results, pairs’ or groups’ outcome scores were significantly higher in the intervention class than in the control class. Most intervention classes scored higher on the task, suggesting that the interventions in this study generally provided effective scaffolding for student problem solving and metacognition to enhance student response outcomes. The most significant improvements in the pair stage were in mathematical performance and expression, while the most significant improvement in the group stage was in ‘Number of the room,’ indicating problem comprehension. Analysis of the cases revealed that both teachers’ interventions may have facilitated students’ understanding of the problem’s meaning. The model both teachers adopted to encourage reflection and discussion was more conducive to students’ problem solving through discussion. The interventions that encouraged students to collaborate may have facilitated their mutual expression, leading students to make better explanations and representations on the task sheets. Among these, Teacher D’s intervention for a more balanced cognitive and social activity was more effective.

This study also found that teacher interventions had a more limited effect on heterogeneous groups. For pairs or groups with different structures, the most effective were high homogeneous groups, followed by low homogeneous groups (homogeneous medium groups among pairs), while there was little difference in performance between the intervention and control classes for heterogeneous groups. The case study revealed that teachers’ individual interventions for the latter students in pairs (or groups) had limited effect, especially for silent, reluctant students, who did not engage in collaboration after the intervention.

10.5.2 Differences in the Level of Control of the Intervention

Both teachers more frequently implemented high-control-level interventions like straitly introduction and explanation to the students (Vermunt & Verloop, 1999) than low-control interventions like hinting or heuristics (Vermunt & Verloop, 1999). This result indicates that both teachers’ intervention styles were biased toward high levels of control.

The two teachers presented different intervention styles when responding to students’ questions, which could be divided into two categories: asking the teacher to explain the questions and asking the teacher to check the students’ results. Both teachers responded less directly, allowing students to make decisions and encouraging them to give ‘reasonable’ and ‘realistic’ answers. The differences between the two teachers were that while Teacher B generally explained the nature of the open-ended tasks, encouraging students to think about them and reflect on them even after completion, Teacher D promoted student discussion, urging the pairs/groups to reach a consensus.

Throughout the student pair and group collaboration stages, Teacher D’s intervention behaviour performance was relatively stable, whereas Teacher B’s showed instability. Regarding intervention characteristics, the frequency, focus, and targets of Teacher B’s interventions differed greatly between the collaboration stages. In the pair stage, Teacher B initiated interventions less often, focused more on cognition, and targeted individuals more while doing the opposite in the group stage.

The two teachers presented different intervention styles, with Teacher B’s interventions being relatively mild and Teacher D’s more strict. Teacher D more frequently initiated interventions than Teacher B. Video analysis showed that Teacher D had a relatively higher level of control in her interventions, was more prepared, systematically targeted some students with direct interventions, and corrected some students’ off-task words, while Teacher B did not. Teacher B always observed students’ performance with a smile on his face and had fewer physical interactions with them, whereas Teacher D would put her hand on students’ shoulders when interacting with them individually to achieve one-to-one instruction, which might have exerted pressure on students.

In Dong et al.’s (2013) study, teacher-initiated instruction was relatively lower, accounting for only 9.9% of collaborative communication. In contrast, in the present study, excluding Teacher B’s performance during the pair collaboration phase, the percentage of teacher-initiated instruction for collaborative communication was significantly higher for both teachers, and students performed better. Teacher D initiated a relatively higher proportion of interventions, and her students performed better, further illustrating the need for teacher instruction in collaborative communication.

The state of the two teachers’ students’ collaboration also differed. Excluding groups that did not fully discuss, Teacher B’s students showed more agreement, while Teacher D’s showed more argument. Previous CPS research has shown that a successful group always features arguments in which group members evaluate others’ perspectives, promote further understanding of the problem, adjust their views, and seek a common solution (Cobb, 1995). Students in Teacher B’s class may have argued less because Teacher B did not adequately encourage it; too little arguing leads group members to suppress or ignore disagreements and form a superficial consensus, resulting in no one challenging wrong ideas. It was observed that students in Teacher B’s classroom were sometimes afraid to express disagreement.

10.5.3 Interventions Based on the Understanding of the Students and Emphasis the Equality

In one part of their interview, teachers were asked about the effectiveness of each group’s learning. In their responses, both teachers paid more attention to students with outstanding performance and the most potential for improvement when evaluating group members’ performance, based on how they understood the students in their daily classroom. Some teacher interventions referenced students’ previous performance, comparing it to their current classroom performance. Van Leeuwen et al. (2015) found similar findings in online collaborative learning cases, where teachers based approximately 35% of their interventions on students’ prior knowledge.

The two teachers had slightly different foci on proactive diagnosis and subsequent interventions. Teacher D focused more on the individual, calling students by name and asking for their thoughts directly in the classroom, whereas Teacher B did not. Teacher B tended to diagnose students’ performance through observation, whereas Teacher D tended to rely more on words. When pairs were not collaborating, Teacher D diagnosed their status and encouraged discussion and communication, whereas Teacher B seemed to focus more on task-related outcomes and less often encouraged discussion. Though both teachers paid special attention to non-collaborating students, Teacher D tended towards direct verbal encouragement, while Teacher B was more observant; he commented on these students’ performance during the interview, confirming his concern for them. Verbal diagnosis is important in facilitating student collaboration. Engeness and Edwards (2017) found that when teachers checked students’ ideas, it helped students summarise and further explain their current progress.

Even though the limited effect of the intervention, the current study finds that both teachers emphasised equality for poorly-behaved pairs and groups. Unlike what has been found in previous studies (Pietarinen et al., 2021; Van Leeuwen et al., 2013), both teachers noticed students with poor academic performance, as confirmed in the interviews. However, in the group stage, teachers preferred interacting with poor discussion groups rather than individuals. Both teachers paid attention to groups that failed to collaborate during the group stage and encouraged them to work on collaborative tasks. However, this encouragement was given only when they observed that half of the group members were not collaborating or the group was still working in pairs. Neither teacher intervened if only one student failed to participate.

10.6 Suggestions

This study has analysed the basic situation and problems in teacher intervention in collaborative problem solving mathematics classrooms and summarised effective teacher intervention strategies. The following strategies provide ideas for effective teacher intervention in collaborative problem solving classrooms, combining the results of the above study with strategies for and discussions of timing interventions and selecting intervention content to overcome existing problems.

Forming Different Types of Collaborative Activities Based on Teaching Objectives

This study found that the two teachers’ diagnoses shared common characteristics, suggesting selective pair (table) or group collaborative learning and collaborative problem solving, depending on the teaching situation. Although there were some differences in their intervention styles, both teachers focused on the individual, with specific interventions in the pair stage, and on the group as a whole in the group collaboration stage, easily ignoring silent group individuals. This is acceptable for a limited time. Given a similar situation in the daily classroom, we suggest that if the task is relatively simple and tends to test individual ability, pair (table) discussion could be considered to facilitate discovering silent individuals (because it is hard to have a discussion if one of the two is silent); if the task is relatively difficult, tends to be more exploratory, and requires more collective wisdom, then group discussion could be considered so the teacher can diagnose overall group performance more intuitively.

It is recommended that teachers effectively choose to conduct different forms of collaboration in daily classroom teaching—table (pair) discussion or group discussion. We suggest forms be chosen based on task difficulty and the collaboration object, in conjunction with the actual teaching situation, to enhance classroom efficiency. Previous studies have found that class and group size significantly impact teacher-student interaction; specifically, it is more likely that spectators or smaller collaborative groups will emerge within oversized groups (Kreijns et al., 2003). A similar phenomenon was found in this study, where groups with more than four students performed relatively worse; in the interview, Teacher D indicated that smaller, more silent collaborative groups existed in groups of six. It can be seen that when designing and conducting collaborative learning and collaborative problem solving, teachers should also arrange and design appropriate group sizes based on class size to maximise collaboration efficiency and achieve instructional goals.

Strategies for Sustained Interaction

Research has shown that most teachers only intervene when they find problems in students’ collaborative processes. While this situation occurs more frequently in classes unfamiliar with collaborative learning, frequent active interventions often interfere with the normal process of student discussions and make students dependent on the teacher. It is recommended that teachers train students’ communication skills through interventions. This can be done through a continuous interaction strategy in which the teacher constantly interacts with the group while participating in the group discussion. The relevant literature and the data in this study identify four strategies teachers commonly adopt when intervening with whole groups: (1) repeating ideas presented by students; (2) asking students to explain their proposed ideas; (3) prompting students to explain the source of their ideas; and (4) encouraging students to compare their respective ideas and reasoning processes.

Encourage students to explain their thoughts through questions like:

  • Can you please explain why you think that way?

  • Can you explain to the group what you mean?

  • Can you please explain how you came to this conclusion?

Follow up with students on their evaluation of others’ ideas through questions like:

  • Could you ask other group members what they think?

  • What do you think of the idea the student just presented? Could you comment on it?

  • Could you please explain why you disagree with the other student’s idea?

  • What is the problem with the other solution?

  • Could you please help him explain his idea?

Group agreement (without precluding outcome diversity) is key to a smooth discussion and successful task completion as a collaborative process. Teachers can use these interventions to guide groups to learn to communicate consistently, keep groups interacting, and get to the root of the problem of frequent teacher-initiated interventions. Long-term training helps students think about these issues spontaneously while engaging in discussion, improving their mathematical communication and collaborative problem solving skills.

Motivational Strategies

Focusing on enlightening intervention content can facilitate the problem solving process. The enlightenment strategy is closely related to how the teacher initiates the intervention and subsequently interacts with students. These are critical moments in the collaborative mathematics problem solving classroom, where the teacher must decide how to intervene and determine the intervention’s focus and content. The data from this study showed that teachers could (1) encourage group members to speak, (2) only observe student discussions (without the teacher speaking), and (3) remind students to read the material carefully. This is a more effective motivational strategy that allows most group members to join the discussion with the teacher’s encouragement; the task-solving process is not ‘contracted’ by a few students, and the teacher only observes and listens to the students’ discussion after initially encouraging them. Even after the teacher leaves, students continue to share their opinions and ideas for a certain time. If students have questions about how to proceed with problem solving, the teacher can help reduce their teacher-dependence by reminding them to read the task materials carefully and review the existing conditions and problems to see where they are having difficulty. However, if students are experiencing difficulties that cannot be solved using pair resources alone, it is important that the teacher promptly provide additional information, which involves information supplementation strategies.

Rule-Making Strategies

In addition to the teacher-initiated strategies used in the intervention process described above, teachers can use rule-making strategies to address the problem of over-frequent teacher-initiated interventions and conduct whole-class interventions before initiating collaborative problem solving. Rule-making strategies involve the long-term development of students’ collaborative problem solving skills. They can be divided into developing rules related to problem solving and developing rules related to collaborative discussions and usually take the form of whole-class teacher interventions. As the researcher did not make this a mandatory aspect of this study, the teachers only explained the task completion rules to their class and did not set rules for the collaborative group discussion process. However, this study found that the lack of discussion rules generally led to conflicts in group work, which led to teacher intervention. As group discussions can be made more efficient by establishing rules, the following suggestions are offered to help teachers develop task resolution rules and discussion rules:

Task resolution rules

  • Clarify the task(s) to be completed

  • Clarify the time needed to complete (each) task

  • Clearly define the participants in the task (e.g., how many people are involved in the discussion) and divide the task among team leaders, reporters, recorders, etc. based on the needs of the activity

  • Clarify the presentation of the task completion results, e.g., filling out task sheets, making posters, etc.

Discussion rules

  • Share your ideas and listen to each other during the discussion

  • When discussing, one person speaks at a time, one after the other

  • Fully explain your ideas

  • Ask ‘why’ if there is a difference of opinion

  • Try to agree on the outcome of the discussion

Once the ground rules have been established in the classroom, they can be used as common-sense guidelines for teachers and students to follow in collaborative problem solving classrooms, facilitating teachers’ efficient guidance and students’ thinking together through verbal communication. After developing these ground rules, collective deliberation (through class meetings and other formats) can ensure they are appropriate and enforceable.

Complementary Information Strategies

Information supplementation strategies involve teacher interventions that offer additional mathematical or task-related information rather than directly providing solutions. Teachers can prompt students to consider the resources available to them. For example, in a house design problem, the teacher could prompt students to consider the size of the classroom bricks and provide the dimensions of the classroom bricks, from which students could estimate the approximate size of each room, enabling them to solve the problem by using available resources. Students may have a greater need for additional information to solve more complex problems. Teachers should be aware of and properly scaffold students’ information needs during interventions.