Keyword

1.1 Introduction

At this time, the socio-economic background of globalization and information technology is presently increasing. Collaboration among team members is crucial to the success of groups, families, corporations, public institutions, organizations, and government agencies (OECD, 2017). Collaboration and communication skills are also considered as indispensable basic skills for future citizens’ overall quality. At all societal levels, breakthroughs on many important issues are often due to teamwork and concerted action and are unable to be solely achieved by individual battles. To help the new generation adapt better to the profound changes in the future society, Chinese governments and international organizations have reportedly ordered the strengthening of basic education, emphasized the importance of inculcating students’ collaborative problem solving abilities, and realized classroom transformation from subject-teaching interdisciplinary integration. In recent years, research has been conducted on teaching collaborative problem solving, with the support of the National Social Science Foundation of China and the Australian Research Council Innovation Fund.

1.1.1 Research Background

UNESCO (1996) proposed that school education in the twenty-first century should focus on four main goals, i.e., studying to learn, perform, unite, and live. This shows that collaboration and communication with others is the basic condition for learning to unite, becoming a new requirement for educating people. To deeply understand the primary skills students should possess in this era, the United States officially launched the 21st-Century Skills Research Project in 2002 and proposed the 4C skills, namely critical thinking and problem solving, communication, collaboration, and creativity and innovation. This proves that collaboration is considered the most important student ability development direction within American education in the twenty-first century (ZHANG, 2012). Furthermore, the Education 2030 project proposed by the Organization for Economic Co-operation and Development (OECD) has attracted much attention in the learning sector, with collaboration and problem solving being included in the predicted 28 competencies. This states that students should participate in teamwork and explore strategies and methods to solve mathematical problems (Cao et al., 2020). A complex and changeable social environment is also responsible for students’ conduct, understanding of basic knowledge, and acquiring disciplinary thinking methods. This implies that the ability to communicate effectively and solve comprehensive problems is an important twenty-first-century school education goal.

The cultivation of collaborative problem solving ability has become more important in curriculum reform and practices in China since the beginning of the century, with the achievement of specific results. However, the collaborative solution ability of 15-year-old Chinese students in Beijing, Shanghai, Jiangsu, and Guangdong was lower than the OECD average (496/500) according to the PISA 2015 Collaboration Problem Solving Report (2017). China was ranked 20th out of 51 countries, with no correlation with reading, science, and math literacy performance (Wang, 2018). Despite this, this sample did not fully represent the country’s overall situation, as the national level was far below these performance stages in many regions. It also reflected the poor ability of mainland students, leading to severe issues for collaborative problem solving learning in China. This was due to the emphasis of the existing traditional teaching methods on teacher explanations while neglecting students’ exploration, specifically the main orientation of subsequent education goals. These observations led to an overemphasis on independent paper and pencil test scores and exam competitions, neglecting peer collaboration and practical innovation. Therefore, teachers only focus on the performance of high-interest exams (high school and college entrance exams), with other factors being closely related.

Based on practice, collaborative problem solving is an important method for cultivating innovative talents. This is due to the implementation of the “Lide and cultivates people” requirements and the development of students’ core literacy. This method has reportedly been considered in Chinese educational research and teaching practices in recent years. On February 23, 2019, the Central Committee of the Chinese Communist Party and the State Council issued Education Modernization of China 2035, based on developing world-class, high-quality learning. This program was launched with Chinese characteristics as an important strategic learning task. It also exhibited the necessity to innovate talent training, implement teaching techniques (heuristic, inquiry, participatory, and collaborative), and focus on cultivating students’ innovative spirit and practical and collaborative abilities (Party and Council, 2019a). The classroom is also the main front of future talent training, indicating the urgency to improve the quality of direct teaching. On June 23, 2019, the Chinese government subsequently issued the Opinions on Deepening Education and Teaching Reform, as well as Comprehensively Improving the Quality of Compulsory Education, explaining the need to optimize teaching methods. It also evaluated the focus on mutual, heuristic, interactive, and inquiry-based teachings, explored the comprehensive learning of subject-based curriculum, and implemented research-based, project-based, and collaborative education (Party and Council, 2019b).

As an important subject in the basic education stage, mathematics plays an important role in cultivating students’ collaborative consciousness and problem solving ability. Since 2000, the evolution of key competencies in the PISA mathematics assessment framework showed that communication has always been an important skill. This states that students must be able to appropriately use symbols, language, etc., to express their thoughts and interact with others. According to the New PISA (2021), communication was one of the eight 21st-Century Skills in the Mathematical Literacy Assessment Framework (Sun et al., 2019). With the reform of the Chinese basic education curriculum, the cultivation of collaborative problem solving ability has gradually penetrated different levels of mathematics education. This requires students to learn the methods of cooperating with others and experience the process of collaborative communication and problem solving. Teachers’ educational concepts are also constantly improved, with students’ learning methods in the classroom being gradually diversified. Besides receptive learning, hands-on practice, independent exploration, and collaborative exchanges have subsequently become important mathematical learning methods, nevertheless, many collaborative teachings are only superficial and mere formalities in the classroom (Wang and Wang, 2022). This is partly because teachers’ organization and implementation strategies are still inadequate due to the lack of collaborative knowledge, even when students’ discussions are found to be very lively (Kirschner et al., 2006). Therefore, more students should participate in the collaborative process and improve learning effectiveness, which has always troubled most front-line mathematics teachers. Although many teachers conceptually recognize this teaching method, it is still rarely used in daily learning due to the pressure of periodical task completion. This explains that teachers only attempt to develop this method for a few minutes when classes open, leading to unsatisfactory and maximal teaching effects (Wang and Wang, 2022).

Based on the international educational background, learning policies, and practical needs, the characteristics of collaborative research and the problem solving process of Chinese middle school students should be deeply investigated. The collaborative problems in real mathematics classrooms should also be analysed, with the methods by which teachers adequately play the roles of educators, guides, and collaborators being explored. To optimize the interaction process in the mathematics classroom environment and cultivate students’ problem solving ability, the National Innovation Fund Project (co-chaired by Professors Cao Yiming and David Clark of Beijing Normal University, China and the University of Melbourne, Australia) launched a program known as “Social Elements in Learning: An Experimental Investigation of Collaborative Problem solving and Knowledge Construction in Chinese-Australian Mathematics Classrooms,” which was approved by the Australian Research Council (ARC) in January 2017. This was based on carrying out the “Empirical Research on Cognitive and Social Interactions, and Their Relationships in Collaborative Problem solving Abilities of Middle School Students.” For secondary school mathematics classrooms, the characteristics of cognitive and social interactions should be explored, with the teaching of collaborative problem solving being analysed. This is based on promoting the cultivation of students’ collaborative problem solving ability and improving the quality of mathematics classroom teaching.

1.1.2 Research Problem

How Did the Learning Activities Occur in the Problem Solving Process?

This problem is the basis of the whole research, with the effective implementation of teaching only being observed after a specific degree of in-depth analysis and knowledge of students’ learning process. This explains that the psychological mechanism of students’ learning process is complex, with problem solving being a high-order thinking and innovative activity affected by many aspects. Much existing research has reportedly been analysed through two aspects, namely cognitive and non-cognitive factors and internal and external variables (Hesse et al., 2015; OECD, 2017). Meanwhile, collaboration is mainly a learning method in teaching scenarios, where students and teachers interact with each other to achieve a common purpose. This process requires the guidance and support of teachers and the participation of families and society. Although the quality of collaborative problem solving depended on the quality of group members’ interactions, much research still proved that not all communications positively impacted student development (Johnson and Johnson, 2009). Teachers’ intervention guidance also has an important impact on student communication (van Leeuwen and Janssen, 2019a, 2019b), with the main question of the research based on the methods by which the social and cognitive interactions were optimized in the collaborative problem solving among middle school students. This cultivates problem solving ability and promotes classroom learning through cognitive and social interactions and teacher guidance. Therefore, the main problem was divided into the following three sub-problems:

  1. (1)

    What are the characteristics of cognitive interaction and knowledge construction in the collaborative problem solving activities of middle school students?

  2. (2)

    What are the characteristics of social interaction and its changing process in the collaborative problem solving activities of middle school students?

  3. (3)

    How do teachers guide students to optimize efficiency and facilitate classroom learning in the collaborative problem solving process?

Based on these research questions, there were two concerned modules, namely theoretical construction and empirical research, which had a total of nine topics as follows:

  1. (1)

    The analytical framework and behavioural process of knowledge construction in collaborative mathematics problem solving for middle school students,

  2. (2)

    The theoretical framework and qualitative research on middle school student’s participation in collaborative mathematics problem solving,

  3. (3)

    The authority relationship of middle school students in collaborative problem solving in mathematics classrooms,

  4. (4)

    The characteristics of mathematical communication in the collaborative problem solving of middle school students,

  5. (5)

    The conflict discourse among secondary school students in collaborative problem solving in mathematics classrooms,

  6. (6)

    Students’ interaction in the middle school’s peer collaborative mathematics problem solving,

  7. (7)

    Teacher Noticing in Collaborative Problem solving in secondary mathematics classrooms,

  8. (8)

    Teacher Intervention in Collaborative Problem Solving in secondary mathematics classrooms,

  9. (9)

    The ability evaluation in Collaborative Problem Solving in secondary mathematics classrooms.

1.2 Literature Review

As the core of human learning and thinking, problem solving symbolizes the cognition of mankind and has reportedly been extended to many disciplines such as science, technology, and engineering. Since the 1980s, team-based problem solving has highly been considered, with national education strongly recommending the utilization of group classroom works for learning activities. Meanwhile, different people have increasingly emphasized sharing and win–win results through teamwork in the twenty-first century, with interpersonal and problem solving skills subsequently obtaining unprecedented attention.

Collaborative problem solving ability is used to effectively participate in team activities, establish common understanding, determine solutions, and obtain collaborative knowledge to solve problems. This ability includes two elements, namely “collaboration” and “problem solving,” as it is also one of the key capabilities necessary for the lifelong development of talents in the twenty-first century. Based on the collaboration level, the social interaction between group members is emphasized, with mutual understanding, team organization, and consensus established, maintained, and achieved through effective communication. However, the problem solving level highlights the cognitive interaction of a task, reflecting the psychological attributes of human beings, including a series of processes such as information extraction, exploration and understanding, and plan execution. In PISA 2015, collaborative problem solving was defined as the ability of an individual to effectively participate in a team of two or more members by sharing understanding, achieving consensus, determining solutions, and uniting collaborative knowledge, skills, and actions to solve problems (OECD, 2017). Based on ATC21S, the team's activities were wholly highlighted, with collaborative problem solving being defined as “a common activity where group members perform a series of steps to complete the transition from a realistic state to an ideal goal” (Zhang et al., 2017). Irrespective of the definition, this ability is still a kind of socialization process, where individuals are found to obtain a high level of cognition through interaction with the social environment. This explains that collaboration and problem solving are not separated, as both are found to often integrate and promote each other. Through the effective collaboration of group members, the complex problem solving transformation process from a realistic state to an ideal goal was also found to be achieved. However, the process is not simply the application of knowledge in a specific discipline or field.

The “input-process-output” model is the basis of group collaborative learning, with the interaction process being closely related to the collaborative effectiveness, as an intermediate link between input and output (Tempelaar, 2004). When the group has external support conditions, its internal interaction process plays an important role in improving teamwork performance. In line with the composition of group members, the collaboration process shows more complicated, multi-dimensional, and dynamic interweaving. Furthermore, the interaction process is more conducive to understanding the essence of group collaboration, for effective guidance towards appropriate promotion (Johnson and Johnson, 2009). Collaborative learning has been widely accepted and drawn attention by academics since 1980s (Bruffee, 1984; Johnson and Johnson, 2009). Decades ago, the shared or situated cognition approach emphasizes social structures where interaction takes place. In this approach, the environment is considered as an integral part of cognitive activities, and knowledge is seen as transferred from one to another, rather, knowledge is constructed through interactions among collaborators (Lai, 2011).

The cognitive interaction research also had a macro description and micro-perspective investigation (Peng and Liu, 2009), with public constructivism stating that knowledge was socially constructed through the collaborative communication between the learners and the social environment. This was due to cognitive support and conflict being considered as two important aspects of psychological interaction. Team members scaffolded each other, argued and negotiated, questioned explanations, understood one another, and constructed interpretations to achieve high levels of cognitive processing and high-quality decisions and practices (Palincsar, 1998). In addition, cognition is the basis of learning, through the exploration of the psychological interaction and advanced thinking process among team members. This had great significance in collaborative problem solving, based on understanding the learning process. This was due to grasping the nature of learning interaction and promoting efficient education.

In the 1990s, the important role of context-based negotiation and renegotiation in constructing classroom knowledge was emphasized, where the research of Vygotsky stated that students’ communication highly promoted the development of individual psychological functions when participating in a task with peers having greater abilities (Chan and Clarke, 2017). This reflected that the process of students’ learning was to realize the construction of knowledge, through interaction with the classroom environment and participation in social activities. Social interaction also plays a key role in knowledge construction and public classroom teaching, as interdependent social interaction between individuals or groups through information dissemination meets several specific needs. The interaction mainly includes five basic elements—subject, carrier, goal, norm, and environment (Zhang, 2012). However, no research has been observed on social interaction in classroom teaching.

Based on the relationship between cognitive and social interaction, previous reports often attached student communication’s social elements to its psychological research elements which subsequently emphasized the complementarity of social and cognitive elements (Cobb and Bowers, 1999), with students opting for the public actions having immediate interaction inducements among group members. In these activities, every student within each group constantly focused on the impact of their behaviours on others and themselves, due to being an important scaffold for reducing cognitive load and promoting classroom knowledge construction. From collaborative problem solving, each involved member expressed personal ideas in their discourse systems, communicated and interacted with other students, and involved the cognitive and social attributes of groups and individuals in the negotiation process. The learning method also focused on the dynamic interplay between problem solving and collaboration, emphasizing the appropriate integration of collaborative social literacy at the individual level. This showed that collaborative problem solving activities supported effective teaching in the classroom, due to being a reliable research hotspot. Presently, few researchers have deeply investigated the classroom field, explored the interactive nature of collaborative problem solving students, and understood the occurrence mode of teamwork learning. During students’ collaborative learning, the essential characteristics and relationship between cognitive and social interaction provided theoretical support for teaching practice and measurement evaluation.

Moreover, teachers’ organization and implementation of group collaboration directly affected students’ participation and interaction effects. According to van Leeuwen (2019), 66 quantitative and qualitative researches on collaborative learning were synthesized to examine the relationship between teachers’ instructional strategies and students’ collaborative processes and effects (Leeuwen and Janssen, 2019a, 2019b). This proved that teachers focused on students’ problem solving strategical feedbacks, helped them plan task progress, and coordinated group collaboration and member participation, positively impacting collaborative processes and results. Therefore, teachers played an important guiding role in group collaborative learning, with proper guidance subsequently promoting the smooth development of collaborative activities. Collaborative teaching also presented different characteristics from traditional classrooms due to their being more complex and teachers’ roles highly multiple. This subsequently led to higher requirements for teachers’ professional ability. Most research on Chinese collaborative learning presently focuses on students, leading to less consideration of teachers’ roles, where a lack of effective empirical analysis has been observed. Based on inefficient collaboration status, there were still many challenges to how teachers can play better roles as educators, organizers, and facilitators.

Cultivating students’ collaborative problem solving ability is an inevitable requirement for implementing the strategy of rejuvenating and strengthening the country through science, technology, and talent. As the core of teaching, the classroom is an important place for cultivating middle school students’ collaborative problem solving ability. This research focuses on secondary school mathematics classroom teaching and performs collaborative learning based on mathematical problem solving. It also deeply explores the nature of teacher–student and student–student communication, optimizes middle school students’ learning in the classroom, evaluates cognitive and social interactions, and creates an autonomous, collaborative, inquiry-based educational environment. In addition, the promotion, innovation, and cultivation of transformation, teaching methods, and students’ collaborative problem solving ability are the core goals of project research and practices.

A combined qualitative and quantitative method was adopted in this research, where collaboratively solvable mathematical tasks were developed to explore the cognitive and social interaction characteristics of collaborative problem solving. This was carried out through several case research and quantitative analyses.

1.3 Methodology

1.3.1 Methods

1.3.1.1 Literature Review and Expert Discussion

By analysing relevant reports in the literature, frontier trends were accurately obtained to determine the plan and questions. Thirty experts with in-depth knowledge of problem solving, collaborative learning, and STEM education were selected, accompanied by the determination of 50 teachers interested in being project participants. Subsequent in-depth evaluations of the problems and analytical frameworks were conducted with the urgency to be solved in collaborative problem solving. This was based on three different backgrounds, namely science, society, and occupation. Suitable mathematical tasks were also designed for middle school students, to carry out collaborative problem solving activities.

1.3.1.2 Classroom Recording

Multi-camera tracking shooting mode was used in the teaching classes, including close-up images of teachers and groups and a panoramic view of students. Two wireless microphones were placed in each group, with teachers utilizing one to ensure clear and complete speech information.

1.3.1.3 Teacher–Student Interviews

Semi-structured interviews were conducted to address the research questions and teachers’ understanding of collaborative problem solving before class activities. After class, students were found to have an in-depth understanding of teachers’ intuitive feelings on teaching, such as their thoughts and reflections on classroom form, collaborative tasks, and students’ performance. Subsequently, students’ first-hand information regarding the collaborative problem solving process (students’ feelings, task-answering situations, etc.) was obtained, which was a valuable supplement to the analysis of video and data.

1.3.1.4 Research Comparison

An in-depth analysis and comparison of the teacher–student, student–student interaction processes were conducted and captured by the video. This showed that the problem solving strategies and methods summarized and extracted the characteristics of cognitive and social interactions, and explored a society for the efficient collaborative solution process.

1.3.2 Research Design

1.3.2.1 The Tasks

With the support of corresponding experts and teachers, we developed a dozen of tasks suitable for collaborative problem solving. Nevertheless, the tasks involved in the current research are mostly open-ended tasks (except Task 3 in the Appendix), which have multiple solutions and methods for the solutions. Open-ended tasks with a low floor to get started and a high ceiling to achievement are suitable for collaboration (Li et al., 2022). The tasks in the current study have symbols and graphic elements consistent with the mathematics curriculum as well as connections with social context.

For example, the “Xiao Ming’s apartment” task led to the following specific problem:

Xiao Ming’s apartment has an area of 60 square metres. There are five rooms in Xiao Ming’s apartment. Draw a possible plan of Xiao Ming’s apartment. Label all rooms and show the dimensions (length and width) of each room.

This task was closely related to the students’ actual lives, requiring them to use existing experiences to solve problems. The problematic task was open-ended and situational and had many possible answers with diverse solution methods. This problem was designed from the geometry content in Chap. 5 in the initial volume of the seventh grade PG (Preliminary Geometry) textbook, People’s Education Edition. Completing the problem required group collaboration of four to six students, with the task being to apply the basic principles of collaborative problem solving design—situational, challenging, ideological, and diverse—to analyse the power of teamwork and strongly support cultivating students’ collaborative problem solving ability (Li and Cao, 2019).

1.3.2.2 Participants

Using purposive sampling, two to four schools representing local average levels were selected in Beijing, Jiangsu, Guangdong, Sichuan, Shanxi, and other provinces/cities. Two to three classes were collected in each school that included a total of 30 classrooms and 1200 students. The sample of teachers has various profiles in gender, age, working experience, and title. All teachers and students have experience in collaborative mathematics learning. Each teacher taught two classes at the same time, and they are similar in size and mathematical achievement on average. All teachers implemented two types of instruction—minimal and structured—in two classes. The structured model provided scaffolding for students’ metacognition skills, although the direct provision of mathematical facts or problem solving procedures differed. The minimal type required little performance from teachers. Under the structured intervention, three forms of teacher–student interaction were permitted: (1) Mathematical, (2) Social-Mathematical, and (3) Social. When teachers evaluated students’ performance, their main goal was to promote students’ working effectively with their peers and groups without directly providing them with the steps to complete the mathematic tasks. This highlighted that students were encouraged to explain and illustrate their ideas and points through diagrams or tables. Besides introducing the task before commencement, teachers should avoid teaching the whole class or assessing the correctness of students’ answers/methods.

1.3.2.3 Role of Researchers

All researchers get permission for collecting and analysing data from headmasters, teachers, and student’s parents. The researchers in this research had no conflict of interest. Although researchers designed tasks and processes of lessons, all teachers could make their own adaptions before and during lessons and express their own ideas and feelings during the interviews.

1.3.3 Data Collection

The process of data collection is shown in Fig. 1.1, which contains five stages. Before formal data collection, the process has been tested and validated in pilot studies. The experts and teachers gave suggestions for optimizing processes after pilot studies.

Fig. 1.1
figure 1

Flow chart of research on collaborative problem solving among middle school students in the classroom environment

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Each teacher in this study carried out the designs to two parallel classes, with minimal or structured teacher interventions. The students’ task list was distributed to the classroom teacher during the preparation stage. It asked teachers to consider and evaluate the procedural steps for categorizing students into groups. Before the lessons, pre-class interviews were conducted to understand teachers’ and students’ familiarity with collaborative problem solving activities and participation. This was accompanied by placing cameras and microphones in appropriate positions, based on the group situation and seat layout (see Fig. 1.2). The camera’s shooting angle is key to capturing students' interaction process in the classroom. Based on students’ seats, each group’s cameras were placed so that all members could be captured without hindering teachers’ normal movement and actions. The classrooms in current study are prepared ahead for capturing teachers’ and students’ interaction by video recording.

Fig. 1.2
figure 2

Schematic diagram of collaborative problem solving teaching grouping (Chan et al., 2018)

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After completion of the preparatory stage, the students’ mathematical collaborative problem solving activities inquiry class was officially implemented. After a few prior plot studies and taking into account teachers’ feedback, researchers worked together to decide on a collaborative problem solving classroom organization process, as shown in Table 1.1. The process contains four sections: Independent thinking, Peer collaboration (two-students groups), Group collaboration (four-students groups), and Summarization. Durations in Table 1 are estimated by previous experience in prior studies, and they are provided for reference to teachers. Teachers were required to carry out designed mathematical collaborative problem solving inquiry courses in their classes.

Audio and video recordings captured students’ interactions and teachers’ behaviours to ensure that the most realistic student–student and teacher–student interaction models became the main focus of data generation. After the course, post-class interviews were conducted with teachers and individual students to understand their feelings, experiences, and strategies regarding this inquiry class. The final data set obtained included all video and audio data of teachers, the group as a unit, the whole class, the recorded teacher and student interviews, and the task lists, including independent thinking and peer and group collaborations.

1.3.4 Data Analysis

Considering that the research questions and focuses varied in different aspects, all authors in the book developed their own methods for data analysis, including analysing task sheets, interactive dialogues within student–student interaction or teacher–student interaction, and interviews in quantitative, qualitative, and mixed-method manners. Multiple software tools, such as NVivo 11, MAXQDA 2018, and SPSS 22.0 were used to restore students’ collaborative problem solving process in the classrooms, comprehend what teachers and students assumed in the collaborative procedure, and ensure valid data analysis. One of the key features of the research is Triangulation, which is a method to increase the credibility and validity of research findings (Cohen & Crabtree, 2008). Through enriched data from different resources, researchers in this research could explore, analyse, and clarify teachers’/students’ behaviour and its reasons.

1.4 The Main Research Results

There are three special focuses in this research: (1) “the path and performance of collaborative knowledge construction for middle school students,” (2) “the social interaction mode in middle school students, based on the collaborative problem solving in mathematics,” and (3) “teachers intervention strategies for middle school students’ collaborative problem solving abilities.” A combined qualitative and quantitative model was designed through video recording, teacher–student interviews, physical collection, and other methods to obtain materials and data from multiple perspectives. The analyses were based on Chinese middle school students’ collaborative problem solving practices and included macro- and micro-level statistical inference analyses to deeply depict the nature of students’ social and cognitive interactions.

1.4.1 Construction of Design-Based Mathematical Collaborative Problem Solving Teaching Model

After repeated evaluations by several peer experts and front-line teachers, 12 suitable mathematical tasks were eventually designed and developed based on three different backgrounds—science, society, and occupation. These mathematical tasks utilized were as follows: (1) open-ended, where multiple solutions enabled different correct answers, (2) symbols and graphic elements, and (3) association with situations outside the classroom.

A design-based teaching model for middle school students was also developed in this research, with a teacher arrangement time in the classroom of approximately 45 min, as shown in Table 1.1. In the class, students completed the prescribed math tasks based on different organizational forms, including independent thinking (one student), peer (two students), and group (four to six students) collaborations. From the collaborative tasks, limited mathematics tools were also allocated to each group, creating and establishing resource interdependence and a positive relationship among the members. For example, a group only provided a task list, without guaranteeing everyone the required tools, so members would have to combine resources to achieve the group goal.

Table 1.1 The organizational arrangement of collaborative problem solving teaching
Full size table

Collaborative learning groups often include four people. When the total number of students in the class is not a multiple of four, individual groups should be selected based on the specific situation. This should be accompanied by increasing the number of people, subsequently allowing groups of approximately six individuals. As the group size increases, the available resources also theoretically elevate to facilitate student achievement. However, the interaction between group members becomes more complex at this time, with students’ collaborative skills being highly required. This leads to the continuous occurrence of problems based on a loss of responsibility and inadequate cohesion. As such, the group size should not be too large. Given the difficulty of the collaboration task in this study, groups of four to six people were appropriate.

1.4.2 Cognitive and Social Interaction During Collaborative Problem Solving in Mathematics Classrooms

We developed our current research on students’ cognitive and social interaction during collaborative problem solving based on previous research. Based on Sun et al. (2020), in-depth research was conducted into the middle school knowledge construction process in collaborative mathematical problem solving, including their collaborative knowledge construction paths, characteristics, levels, and performances (Sun, 2020). The results stated that the students’ overall participation in collaborative knowledge construction was good, due to its being the main focus of mathematical problem solving. They also experienced consensus formation, information sharing and understanding, disagreement discovery and clarification, content negotiation and co-construction, and verification and adjustment. Six stages were observed in the evolution of multiple perspectives, including integration, fission, mutation, etc. Based on these results, a new perspective was provided to comprehend students’ collaborative learning quality and improve the efficiency and quality of collaborative problem solving.

Core social interaction issues in students’ collaboration process for mathematical problem solving were subsequently explored, such as (1) the characteristics of peer collaboration and interaction, (2) mathematical communication, (3) authoritative relationship, (4) the type and structure of conflict discourse, and (5) the process of learners’ participation. These issues were based on producing in students’ collaborative learning the elements and characteristics of social interaction needed to optimize the communication process and cultivate students’ collaborative ability. Based on Zhang et al. (2021), a positioning theory was initially used to construct a framework for middle school students’ participation in collaborative problem solving. A “negotiation event coding and chain analyses-interactive role position coding and change process-story line construction” was also developed as the main experimental path (Zhang et al., 2021). This explored group members’ interaction roles and the evolution of the negotiation topic during the collaboration process based on micro-case research. The results showed that student’s participation in the negotiation events included initiation, response, evaluation, non-interaction, and non-speaking, and that their role patterns changed. These were divided into three role models—single, combined, and transformed—that provided theoretical guidance for understanding students’ social participation path within the collaborative problem solving process. Several experts have conducted empirical research on teachers’ intervention strategies in students’ collaboration process.

Optimizing middle school students’ cognitive and social interaction has become a hot topic in collaborative learning research. This study, based on secondary school mathematics classrooms, emerged from the multidisciplinary cognitive and social psychology perspectives. The core topic of collaborative problem solving was evaluated using empirical methods, including (i) the mathematical collaborative teaching model, (ii) the path and performance of collaborative knowledge construction for middle school students, and (iii) the model and characteristics of social interactions. Moreover, an in-depth analysis of teachers' role in student collaboration was carried out to excavate interactive elements, help achieve efficient mathematical teaching, and improve middle school students’ collaborative problem solving ability.

In the series of research on cognitive and social interaction, the authors focus on topics including collaborative knowledge building (Chap. 3), negotiation discourse (Chap. 4), social authority (Chap. 5), opportunity to learn (Chap. 6), mathematical communication (Chap. 7), conflict discourse (Chap. 8), and peer student interaction (Chap. 9).

1.4.3 Teacher Noticing and Guidance During Collaborative Problem Solving

The current research also focuses on the teacher’s role and behaviour in collaborative problem solving, which has attracted broad interest (Webb, 2009; van Leeuwen and Janssen, 2019a, 2019b). Based on (Dong et al., 2013), a case study was conducted on teachers’ intervention activities in secondary school mathematical collaborative learning classrooms. This depicted that Chinese mathematics teachers mainly judged the group process through observation, often aiming their intervention objects at individuals rather than the entire group. Their intervention content also lacked guidance and evaluation of collaboration and communication, which provide many inspirations for and reflections on collaborative teaching in Chinese mathematics classrooms. To have a deeper understanding of teachers’ assessment of groups, (Li et al., 2021) used eye-tracking technology to analyse the attention of pre-service mathematics educators. The results confirmed that pre-service teachers focused on students’ problem solving ideas and outcomes, due to their being attracted to learners who spoke most frequently. However, sufficient focus was not fixed on those with low participation. The analysis of mathematics teachers’ attention to students’ collaboration processes also helped effectively explain educators’ behavioural intentions and their causes from a cognitive psychology perspective.

Experienced-pre-service comparison of teacher noticing during students’ collaborative problem solving is done through eye-tracking in Chap. 10, where find that experienced teachers distribute their attention more evenly and notice more important facets of group teaching. Teacher intervention during collaborative problem solving is evaluated and analysed in Chap. 11, where find teacher intervention is mostly effective, but less effective for heterogeneous groups. Authors in Chap. 11 also discussed control and equality of teacher intervention and suggestions are also given for fostering teacher guidance during collaborative activities. In Chapter 12, the author evaluates students’ ability to collaborative problem solving, finding that students lack a sense of collaboration in communicative dialogue, giving suggestions on training ability.