“Anything taking shape?” : Capturing various layers of small group collaborative problem solving in an experiential geometry course in initial teacher education

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ “Anything taking shape?” : Capturing various layers of small group collaborative problem solving in an experiential geometry course in initial teacher education © The Author(s) 2021 Published version Pöysä-Tarhonen, Johanna; Häkkinen, Päivi; Tarhonen, Pasi; Näykki, Piia; Järvelä, Sanna Pöysä-Tarhonen, J., Häkkinen, P., Tarhonen, P., Näykki, P., & Järvelä, S. (2022). “Anything taking shape?” : Capturing various layers of small group collaborative problem solving in an experiential geometry course in initial teacher education. Instructional science, 50(1), 1-34. https://doi.org/10.1007/s11251-021-09562-5 2022 Vol.:(0123456789) Instructional Science https://doi.org/10.1007/s11251-021-09562-5 1 3 ORIGINAL RESEARCH “Anything taking shape?” Capturing various layers ofsmall group collaborative problem solving inanexperiential geometry course ininitial teacher education JohannaPöysä‑Tarhonen1 · PäiviHäkkinen1 · PasiTarhonen2· PiiaNäykki3,4 · SannaJärvelä4 Received: 13 May 2020 / Accepted: 17 October 2021 © The Author(s) 2021 Abstract Collaborative problem solving (CPS) is widely recognized as a prominent 21st-century skill to be mastered. Until recently, research on CPS has often focused on problem solution by the individual; the interest in investigating how the theorized problem-solving constructs function as broader social units, such as pairs or small groups, is relatively recent. Capturing the complexity of CPS processes in group-level interaction is challenging. Therefore, a method of analysis capturing various layers of CPS was developed that aimed for a deeper understanding of CPS as a small-group enactment. In the study, small groups of teacher education students worked on two variations of open-ended CPS tasks—a technology-enhanced task and a task using physical objects. The method, relying on video data, encompassed triangulation of analysis methods and combined the following: (a) directed content analysis of the actualized CPS in groups, (b) process analysis and visualizations, and (c) qualitative cases. Content analysis did not show a large variation in how CPS was actualized in the groups or tasks for either case, whereas process analysis revealed both group- and task-related differences in accordance with the interchange of CPS elements. The qualitative cases exemplified the interaction diversity in the quality of coordination and students’ equal participation in groups. It was concluded that combining different methods gives access to various layers of CPS; moreover, it can contribute to a deeper articulation of the CPS as a group-level construct, providing divergent ways to understand CPS in this context. Keywords Collaborative problem solving· Directed content analysis· Interaction· Method triangulation· Process analysis· Process visualizations * Johanna Pöysä-Tarhonen johanna.po[email protected] Extended author information available on the last page of the article J.Pöysä-Tarhonen et al. 1 3 Introduction This paper presents the development of a method capturing various layers of collaborative problem solving (CPS) for a deeper understanding of CPS as a small-group enactment. CPS has received substantial interest as one of the key competencies of 21st-century learners in educational reforms of national curricula worldwide and international, largescale assessments (i.e., assessment and teaching of 21st Century skills ATC21S, http:// www. atc21s. org; the Organisation for Economic Co-operation and Development OECD, Programme for International Student Assessment PISA 2015, http:// www. oecd. org/ pisa/). Because of the origins of CPS constructs, which stem from assessing individual problem solving, until recently, research on CPS has often focused on problem solving and the skills individual learners bring to the joint problem space. Against this background, an interest in investigating how the theorized CPS constructs function as broader social units, such as small groups, is relatively recent (Dowell etal., 2020; Funke etal., 2018; Graesser etal., 2018; Scoular etal., 2017). CPS is a complex and dynamic construct, where the “complexity label” is not based on cognitive demands but refers to the definition of the CPS comprising multiple interacting elements (Scoular etal., 2017). Grounded in the socio-cognitive approach to learning, CPS lies in a two-dimensional space of social and cognitive domains that are tightly coupled and intermingle in the problem-solving process (e.g., Avry etal., 2020; Care etal., 2016; Dowell etal., 2020; Hesse etal., 2015; Swiecki etal., 2020). Given the social nature of CPS, the socio-cognitive properties reside and evolve in interaction between the problem solvers (Dowell etal., 2020). Yet, because of the inherent complexity of the construct, determining how to identify the manifested behaviors of CPS and capture the processes at the group level is challenging (Dowell etal., 2020), especially in open-ended problem spaces that lack clarity of the problem solution (Scoular etal., 2017). Despite their usefulness in certain situations, methods that rely primarily on participants’ perceptions of collaboration and CPS via questionnaires, self- or peer evaluations (for an overview, see Kyllonen etal., 2018), or analyzing the contents of communication—irrespective of their semantic and temporal relations—can afford limited understanding around the actual processes of collaboration (Dowell etal., 2020; Swiecki etal., 2020). The CPS construct and its social dynamics have been simulated in structured, computerbased environments that automatically generate large-scale data samples (e.g., Dowell etal., 2020; Graesser etal., 2018; Swiecki etal., 2020). These data allow for analysis based on computational linguistics frameworks (i.e., group communication analysis, GCA; Dowell etal., 2018, 2020) that quantify, for example, social dynamics in groups involved in CPS. However, together with simulated CPS activities and automated analysis at a large scale, searching for ecologically valid measures based on detailed capturing of the microinteraction processes at the small-group level (e.g., Davis etal., 2015)—in authentic learning contexts that large-scale studies can also account for (Reimann, 2021)—is similarly desired (Gauvain, 2018). Consequently, if CPS processes are seen to consist of intertwined and interdependent interactions among the group that evolve over time, adequate methods are needed that allow for detailed capturing and analysis of these interaction-related events (Reimann, 2021) and the complex mechanisms of “coupling minds” (Cress etal., 2018) involved in CPS in this context. In this study, to better understand CPS as a group-level enactment, a method is developed for capturing various layers of CPS by relying on triangulating analysis techniques (Humble, 2009; Meadows & Morse, 2001). In triangulation, at least two analysis “Anything taking shape?” Capturing various layers ofsmall… 1 3 approaches are integrated to form a deeper picture of the object of investigation (Flick, 2004; Kelle & Erzberger, 2004). In this study, triangulation combines the following: (a) directed content analysis on the actualized CPS in groups (e.g., Hsieh & Shannon, 2005; Humble, 2009), (b) process analysis and visualizations (e.g., Abbott, 1990; Kapur, 2011; Reimann, 2009, 2021), and (c) qualitative cases (e.g., Davis etal., 2015; Stahl, 2017). All the analyses are based on the communication stream as video recordings from CPS sessions in small groups (Barron etal., 2015). In the study, small groups of teacher education students solve open-ended CPS tasks during an experiential mathematics education course in geometry. The tasks include a technology-enhanced task using dynamic software, and a task using physical objects. In a colocated CPS, participants work together in the same physical location and coordinate their understanding and actions by exchanging ideas, knowledge, or resources to collaboratively converge on the correct understanding of the shared problem (Alterman & Harsch, 2017). However, groups often fail to make use of their full potential (Barron, 2000, 2003). Therefore, to better ensure that productive CPS processes evolve, the pedagogical design in this study incorporates the task characteristics (as sufficient complexity of the task) to enhance the process gains of collaboration (Sears & Reagin, 2013) and includes external support as a macro-script, an activity model to engage learners in group interaction processes (e.g., Dillenbourg & Hong, 2008). However, because this paper emphasizes the methodological aspects of capturing various layers of CPS as a small-group enactment, the scope does not cover how the pedagogical design affects the quality of CPS. Next, the two focal points of this study, the CPS construct and the method, are described in more detail. Collaborative problem solving: aconstruct definition In CPS, collaboration and problem solving intermingle, requiring both the social and cognitive contributions of the participants (Care etal., 2016; Funke etal., 2018; Hesse etal., 2015; Scoular & Care, 2020; Scoular etal., 2017). Basically, the social components of CPS are related to how participants coordinate and communicate with one another (e.g., Richardson etal., 2007), as well as how they regulate and resolve differences among the collaborating partners (e.g., Järvelä etal., 2016). The cognitive elements, in turn, are related to how effectively and efficiently participants solve the problem (e.g., Mayer, 1998). To be successful, CPS requires engagement in shared processes and exchanges of ideas, knowledge, or resources, as well as externalizing understandings and efforts in the group. In observed group interaction, CPS is actualized using verbal and nonverbal indications and constant updates that are transparent to most group members (Graesser etal., 2018). This means that for the group to be aware of most of the substantial problem parts and to work successfully and reach a mutual understanding, the participants are obliged to communicate, exchange, and share knowledge over problem-solving processes at the social and cognitive levels (Graesser etal., 2018; Hesse etal., 2015). This requires a willingness to collaborate and share expertise, as well as the ability to manage interpersonal conflicts (Hesse etal., 2015). To be productive, CPS not only makes use of the full potential of the group’s expertise but also necessitates the full set of social skills coming into force. In this study, for detailed articulation of the CPS construct in small groups of learners, a comprehensive CPS framework by Hesse etal., (2015; see also Care etal., 2016; Scoular & Care, 2020; Scoular etal., 2017), originally developed for the assessment of CPS skills in the ATC21S project, is applied (Care etal., 2016; Griffin & Care, 2015). The framework is chosen because it profoundly covers both social and cognitive elements of J.Pöysä-Tarhonen et al. 1 3 the CPS construct (cognitive, social, and regulatory aspects), and it amalgamates existing theoretical knowledge from social psychology and problem solving (Hesse etal., 2015). Expanding from the original usage, the aim is to explore how the CPS elements intermingle and interact in the group-level processes and over time. The framework gives a detailed description of how the social and cognitive elements of the CPS construct can come into force in a collaborative context (see Appendix 1), and therefore, it is suitable for the purposes of directed content analysis (Hsieh & Shannon, 2005) as employed in this study. In the framework, CPS is defined as a set of elements consisting of “social” and “cognitive” as the first-order components (see Table1). As the second-order components, the framework covers three social process elements (participation, perspective taking, social regulation) and two cognitive process elements (task regulation, knowledge building). The five elements are further divided into 19 specific sub-elements as third-order components. From the social set, “participation” (sub-elements: action by individual and interaction with the partner or partners) refers to the willingness and eagerness to share information, externalize one’s thoughts, and be involved in the different stages of problem solving. “Perspective taking” (Hayashi, 2018; sub-elements: adaptive responsiveness, audience awareness) means the ability to understand and consider others’ perspectives and contributions and to tailor one’s contributions to others. “Social regulation” points to the knowledge and awareness of the strengths and weaknesses of the other group members (sub-elements: transactive memory, Lewis & Herndon, 2011; negotiation, Thompson etal., 2010; selfevaluation [metamemory], Wegner, 1986; and responsibility initiative, which signifies taking responsibility for the progress of [the parts of] the task by the group). From the cognitive set, “task regulation” refers to planning and monitoring skills for developing strategies for problem solving and shared problem representation (“joint problem space”; Roschelle & Teasley, 1995). The sub-elements of task regulation incorporate problem analysis, setting goals, resource management, tolerance for ambiguity, collection of information, and systematicity as four aspects of planning, one of the fundamental activities in CPS (Eichmann etal., 2019). Moreover, from the cognitive set, “knowledge building” refers to the ability to learn and build knowledge through group interaction (Scardamalia & Bereiter, 2006). In CPS, steps in knowledge building include the sub-elements of identification of relationships, cause and effect, and testing of hypotheses (Griffin, 2014). Taken together, these five CPS elements and their 19 sub-elements are considered essential for successful CPS activity. In the framework, CPS may not proceed linearly; rather, the CPS elements can overlap and run parallel between the different stages of the process. Applying analysis method triangulation forcapturing various layers ofCPS asagroup‑level enactment The concept of triangulation (e.g., Flick, 2002, 2004; Kelle & Erzberger, 2004) has multiple meanings in the literature, but it is generally viewed as a cumulative way of validating the results or as “an enlargement of perspectives that permit a fuller treatment, description and explanation of the subject area” (Kelle & Erzberger, 2004, p. 174). Here, the emphasis is on the latter purpose—to increase the scope, depth, and consistency of research (Flick, 2002). The same problem can be approached using various methods; alternatively, triangulation can be employed to treat different aspects of the same phenomenon (Kelle & Erzberger, 2004). In this study, three different methods of analysis (directed content analysis, process analysis and visualizations, and case examples) are used to capture various layers of CPS as a group-level enactment. All the methods are valued equally and can add to a “Anything taking shape?” Capturing various layers ofsmall… 1 3 Table 1 Collaborative problem solving (CPS) construct definition Adapted from Care etal. (2016), Hesse etal. (2015), Scoular and Care (2020), Scoular etal. (2017) CPS elements First-order components Social skill set: “Collaborative” aspects of CPS Cognitive skill set: “Problem-solving” aspects of CPS Second-order components Participation Perspective taking Social regulation Task regulation Knowledge building Third-order components Action, interaction, task completion Adaptive responsiveness, audience awareness Self-evaluation, transactive memory, responsibility initiative Problem analysis, setting goals, resource management, flexibility and ambiguity, collecting elements of information, systematicity Relationships, “if…then” rules (cause and effect), “what if” hypothesis solution J.Pöysä-Tarhonen et al. 1 3 unified depiction in understanding CPS in this context. Next, the set of methods chosen is discussed and presented with examples of related analysis approaches. Using a directed approach (e.g., Hsieh & Shannon, 2005; Humble, 2009), content analysis is a deductive category application. It is guided by a structured process where an existing theory or conceptual frame is applied to a novel context, and the categories and the determined operational definitions of each category are based on the theory or framework (Hsieh & Shannon, 2005). In the current work, directed content analysis applies the ATC21S CPS framework and the codebook by Hesse etal. (2015) in analyzing interactional data. The evidence from the coding can be presented, for example, by offering descriptive evidence, or in this study, first calculating the frequency distributions of the categories that enter into the broader framework (Vogel & Weinberger, 2018). A related approach, quantitative content analysis (e.g., Neuendorf, 2011), also referred to as the “coding and counting approach” in analyzing interaction, is a summarizing, quantitative analysis (Jeong et al., 2014); in contrast to directed content analysis, it applies inductive coding (Kennedy, 2018). In quantitative content analysis, a coding scheme is developed and applied in the data corpus (De Wever etal., 2006). When studying collaborative learning and CPS, the analysis has typically resulted in frequency counts of different categories as a means of comparing different experimental conditions (i.e., concerning the processes or outcomes; Csanadi etal., 2018; Swiecki etal., 2020). “Coding and counting” analyses are seen as useful in explaining the distribution of process categories, that is, regarding the variations of the outcomes (Kapur, 2011; Reimann, 2009). However, when frequency-based methods are applied in studies of collaborative learning and CPS, it has been found, for example, that groups with similar frequency distribution of categories may well have different temporal dynamics of these categories (Kapur, 2011). As Hesse etal. (2015) stated, CPS processes unfold over time and can vary over the course of problem solving. Therefore, to avoid treating CPS as a set of isolated events, an issue that cumulative accounts or frequencies are often criticized for (e.g., Csanadi etal., 2018; Kapur, 2011; Reimann, 2009; Swiecki etal., 2020), the temporal dimension of the data is also acknowledged. As Kapur (2011) pointed out, “learning in general, and problem solving in particular, is a continuous, dynamic process that evolves over time” (p. 39). In the current work, studying temporality in collaborative learning and CPS processes is inspired by the previous work of several authors (e.g., Avry etal., 2020; Csanadi etal., 2018; Kapur, 2011; Reimann, 2009, 2021; Reimann et al., 2011; Stahl, 2017; Swiecki etal., 2020). Various methods are applied for analyzing processes and temporality in collaborative learning and CPS studies (for an overview, see Lämsä etal., 2021a; Reimann, 2009, 2021). For example, in epistemic network analysis (ENA), which is a discourse analysis technique that models the temporal co-occurrences of codes in discourse, temporality is studied by focusing on the recent temporal context and immediate events of CPS in the groups (e.g., Andrist etal., 2018; Csanadi etal., 2018; Swiecki etal., 2020; Williamson Shaeffer, 2017). Lag-sequential analysis (LsA; e.g., Bakeman & Gottman, 1997; Bakeman & Quera, 2011) has been employed for modeling CPS processes using transitional probabilities between sequencies of collaborative action; in this way, researchers have demonstrated, for example, the significant relationship between variation in temporal patterns and variation in group performance (Kapur, 2011). A related sequential analysis approach, sequential pattern mining (e.g., Febrer-Hernández & Hernández-Palancar, 2012), has been applied, for example, for modeling students’ ill-defined problem solving (Norm Lien etal., 2020). Both methods model sequences, but if compared to LsA, sequential pattern mining has “Anything taking shape?” Capturing various layers ofsmall… 1 3 been used to mine sequences of events that are frequent (i.e., occur more often than a minimal level of frequency) in a dataset (e.g., Chen etal., 2017). Temporality not only comes into play in quantitative terms (i.e., duration) but also has significance in terms of the order of events (Reimann, 2009, 2021). In the current study, the indicators of temporal events were the order and intensity of the social and cognitive states and state changes during task completion. The social and cognitive states refer to the coded, timestamped (social and cognitive) sub-elements of CPS (see Appendix 2), which are amalgamated back into broader social and cognitive strands (i.e., the first-order components of the CPS construct). The intensity of the states is related to how many of the different sub-elements of the social or cognitive strands appear at the same point in the CPS session. The state changes are based on calculations of inter-event distances as temporal distances between social states and cognitive states over the CPS session, whereas the calculated averages of states and state changes cover the entire CPS session (e.g., Abbott, 1990; see also Reimann, 2009); all are identified from the timestamped coded interaction of the groups. Accordingly, in this study, the indicators—the order and intensity of the social and cognitive states and the (averages of) states and state changes—are expected to pinpoint the interchange of joint efforts and related communication with the partners, which form the basis of collaboration and CPS (e.g., Dillenbourg etal., 2016; Roschelle & Teasley, 1995). Finally, via qualitative cases, attention is turned back to the micro-interaction processes in the groups (Davis etal., 2015; Stahl, 2017). All groups are unique in terms of how the actions and interactions come together to form the shared practice (Stahl, 2017). In addition, case examples can provide intensification and offer explanations to help in the interpretation of the previous phases of analysis (Flick etal., 2004). Here, qualitative cases can show whether the differences in the temporal occurrence of CPS from the previous phase of analysis are visible at the micro-interaction level. In general, when carefully justified, cases can provide a thorough view of a complex social phenomenon (Baškarada, 2014). Cases do not aim to generalize to populations, but they can arrive, for example, at conceptualizing the regularities of small-group processes in CPS (Stahl, 2017). Aims This study applies analysis method triangulation to capture various layers of CPS enactment in small, co-located groups during a mathematics education course in teacher education. The research questions are as follows: RQ1: How are the theorized CPS construct (comprising social and cognitive elements and their sub-elements) actualized in the groups’ interactions in two different task designs (i.e., technology-enhanced vs. using physical objects), interpreted as cumulative accounts? RQ2: How do the elements of the CPS construct vary over time during problem solving in the different groups and different tasks? What differences are there in temporal occurrence in this regard? RQ3: In what ways are the differences in the temporal occurrence of CPS elements visible in the qualities of interaction in the groups (exemplified as case examples)? J.Pöysä-Tarhonen et al. 1 3 Table 2 Descriptions of the Collaborative Problem Solving (CPS) Tasks (Sessions 1 and 2) CPS Session 1: Tessellations with GeoGebra software CPS Session 2: Giant tessellations with physical objects (cardboard tiles) Task 1: Covering the plane Using GeoGebra, design a pattern that covers the plane. Describe the pattern in terms of translations, rotations, and reflections. Design another pattern. Describe the pattern in terms of translations, rotations, and reflections. A pattern is said to be symmetric if the whole pattern can be translated, rotated, or reflected so that it appears identical. Which symmetries exist in the pattern you designed? Task 2: Covering the plane—a sequel Select a pattern and make it from the physical shapes. Use the colors as you like. Illustrate the used translations, rotations, and reflections with the physical shapes. “Anything taking shape?” Capturing various layers ofsmall… 1 3 The dark gray color points to the social states of CPS skills, whereas the light gray color points to the cognitive states. The three areas marked with the transparent, lightest gray color represent the timepoints at which the script cards were discussed. The intensity of the appearance, reflecting the number of social or cognitive sub-elements activated at a specific point, is visualized by the height of the “ray.” Phase 3: qualitative cases based onselected interactions Based on the results of the previous phase (i.e., the proportional differences between the averages of state changes in the two tasks), in Phase 3, the focus returned to the interaction data to better understand the group-level differences in this regard. In the analysis, the aim was to concentrate on identifying the characteristics of interaction when the groups were completing the second task and searching for chains of events that would exemplify the differences. Fig. 4 Section from the coded excel data file (Task 1, the onset of the task). The coded data file includes the start and end timestamp of the coded interaction, the actor and the (Atlas.ti) id of the coded fragment, the coded seven collaborative problem solving (CPS) sub-elements, the timestamp running from the beginning of the video, the values for cognitive and social states (COGNITIVE, SOCIAL) as well as cognitive state changes (CTRAVEL) and social state changes (STRAVEL) Fig. 5 Example of a CPS circle. The dark gray color points to the social states of CPS skills, while the light gray color points to the cognitive states. The three areas marked with the transparent, lightest gray color represent the timepoint at which the script cards were discussed. The intensity of the appearance, reflecting the number of social or cognitive sub-elements activated at the specific point, is visualized with the height of the “ray.” J.Pöysä-Tarhonen et al. 1 3 Results RQ1: How did thetheorized collaborative problem‑solving elements actualize inthegroups’ interactions? In general, the cumulative accounts did not indicate a large variation between the different groups in either task design. When focusing on the sub-elements of the CPS, Problem Reanalysis from the cognitive set formed the most frequent sub-element in both the technology-enhanced task (see Table7) and the task utilizing physical objects (see Table8). In Task 1, the Participation sub-element from the social set was the second largest sub-ele- ment, whereas Task Exploration from the cognitive set was the third largest sub-element, except in Group D, where Task Exploration formed the second largest element from all the sub-elements, followed by Participation as the third largest sub-element. In Task 2, the largest element, Problem Reanalysis, was followed by Participation as the second largest and Coordination from the cognitive set as the third largest sub-element. In both task designs, it seems that cognitive elements were more identifiable from the interaction data but with slightly different task-related emphases: Task 1 involved more task exploration than coordination, while Task 2 involved more coordination than task exploration. From the descriptive statistics, when focusing on the dispersion of the CPS sub-ele- ments, the low SD implies a more or less equal appearance of CPS elements in the different groups and task designs, as indicated in Tables9 and 10. The measures of central tendency (average, median) indicate the same result. Table 7 Proportions of different collaborative problem solving (CPS) elements in different groups (Task 1, Technology- Enhanced Task Design) Group Proportion Proportion Proportion B (%) C (%) D (%) CPS sub-element Participation 17 21 22 Perspective Taking A 7 8 7 Perspective Taking B 10 11 9 Coordination 15 10 11 Task Exploration 16 15 24 Problem Analysis 5 6 7 Problem Reanalysis 49 59 55 Table 8 Proportions of different collaborative problem solving (CPS) elements in different groups (Task 2, Task Design Utilizing Physical Objects) Group Proportion Proportion Proportion B (%) C (%) D (%) CPS sub-element Participation 23 31 22 Perspective taking A 6 5 9 Perspective taking B 7 13 14 Coordination 16 19 17 Task exploration 7 8 9 Problem analysis 11 8 9 Problem reanalysis 52 52 48 “Anything taking shape?” Capturing various layers ofsmall… 1 3 RQ2: How did thecollaborative problem‑solving elements vary overtime? Because the results from the first phase of analysis did not show a large variation in different groups or tasks, the temporal information of these coded data was elaborated further. Here, it was asked how the elements of CPS varied over time by focusing on the order and intensity of social and cognitive states (Figs.6, 7, 8, 9, 10, 11) and comparing the (average) alignment of the social and cognitive states and state changes in the groups regarding the two different tasks (Table11). The results suggest that in Group B, the average occurrence of social states is slightly higher in Task 2 than it is in Task 1 (Task 1: 0.115, Task 2: 0.122), but the value for state changes regarding social elements is remarkably lower in Task 2 compared with Task 1 (see Table11). (The calculated proportional difference between the tasks is 35.2% less). It seems that at the group performance level, more social states appeared in the task with the physical objects than with GeoGebra, but the continued “social moments” in the group interaction, on average, were longer. Figures6 and 7 visualize the actualized interchange of the social and cognitive states in Group B. For Group C (see Table11), there was an increase in the average occurrence of social states in Task 2 compared with Task 1 (Task 1: 0.135, Task 2: 0.164). In addition, the value for the average of social state changes was higher in Task 2 (using physical objects) compared with technology-enhanced Task 1. (For Task 2, the calculated proportional difference is 10.7% more in this regard). This indicates that although more social states occurred on average, the social moments were slightly shorter in Task 2 compared with Task 1. Table 9 Standard deviation (SD), average, and median (M) of the collaborative problem solving (CPS) elements in task 1 (Technology-Enhanced Task Design) Group SD Average Median B, C, D (%) B, C, D (%) B, C, D (%) CPS sub-element Participation 2 20 21 Perspective taking A 1 7 7 Perspective taking B 1 10 10 Coordination 2 12 11 Task exploration 4 18 16 Problem analysis 1 6 6 Problem reanalysis 4 54 55 Table 10 Standard deviation (SD), average, and median (M) of the collaborative problem solving (CPS) elements in task 2 (Task Design Utilizing Physical Objects) Group SD Average Median B, C, D (%) B, C, D (%) B, C, D (%) CPS sub-element Participation 4 25 23 Perspective taking A 2 7 6 Perspective taking B 3 12 13 Coordination 1 17 17 Task exploration 1 8 8 Problem analysis 2 9 9 Problem reanalysis 2 51 52 J.Pöysä-Tarhonen et al. 1 3 Figures8 and 9 visualize the actualized interchange of the social and cognitive states in Group C. In Group D (Table11), there was a decrease in the averages of the cognitive states (Task 1: 0.246, Task 2: 0.208), as well as a decrease in the average of the cognitive state changes in Task 2. The calculated proportional difference between Tasks 1 and 2 was 24.6% less in this regard. This means that, in Task 2, fewer “cognitive moments” occurred, but on -0.33 0.00 0.33 0.66 0.99 -0.25 0.00 0.25 0.50 0.75 1.00 Fig. 6 Collaborative problem solving (CPS) circles from Group B in Task 1 (technology-enhanced task). The figure on the left (dark grey color) presents social states and state changes, the figure on the right (light grey color) cognitive states and state changes, the script discussions are marked with the lightest grey color -0.33 0.00 0.33 0.66 0.99 -0.25 0.00 0.25 0.50 0.75 1.00 Fig. 7 Collaborative problem solving (CPS) circles from Group C in Task 1 (technology-enhanced task). The figure on the left (dark grey color) presents social states and state changes, the figure on the right (light grey color) cognitive states and state changes, the script discussions are marked with the lightest grey color “Anything taking shape?” Capturing various layers ofsmall… 1 3 average, these moments lasted longer than they did in Task 1. Figures10 and 11 visualize the actualized interchange of the social and cognitive states in Group D. -0.33 0.00 0.33 0.66 0.99 -0.25 0.00 0.25 0.50 0.75 1.00 Fig. 8 Collaborative problem solving (CPS) circles from Group D in Task 1 (technology-enhanced task). The figure on the left (dark grey color) presents social states and state changes, the figure on the right (light grey color) cognitive states and state changes, the script discussions are marked with the lightest grey color -0.33 0.00 0.33 0.66 0.99 -0.25 0.00 0.25 0.50 0.75 1.00 Fig. 9 Collaborative problem solving (CPS) circles from Group B in Task 2 (task design utilizing physical objects). The figure on the left (dark grey color) presents social states and state changes, the figure on the right (light grey color) cognitive states and state changes, the script discussions are marked with the lightest grey color J.Pöysä-Tarhonen et al. 1 3 RQ3: In what ways were thedifferences inthetemporal occurrence ofCPS elements visible inthequalities ofinteraction inthegroups? In the third phase, to search for explanations for the results from the second phase of analysis, the focus returned to exploring the interaction processes in the groups. With the case examples, as “a sensible follow-up” to the previous levels of analysis (Flick etal., 2004), -0.33 0.00 0.33 0.66 0.99 -0.25 0.00 0.25 0.50 0.75 1.00 Fig. 10 Collaborative problem solving (CPS) circles from Group C in Task 2 (task design utilizing physical objects). The figure on the left (dark grey color) presents social states and state changes, the figure on the right (light grey color) cognitive states and state changes, the script discussions are marked with the lightest grey color -0.33 0.00 0.33 0.66 0.99 -0.25 0.00 0.25 0.50 0.75 1.00 Fig. 11 Collaborative problem solving (CPS) circles from Group D in Task 2 (task design utilizing physical objects). The figure on the left (dark grey color) presents social states and state changes, the figure on the right (light grey color) cognitive states and state changes, the script discussions are marked with the lightest grey color “Anything taking shape?” Capturing various layers ofsmall… 1 3 the interest was in seeking clarifications for the proportional, group-related differences between the averages of the state changes regarding the two task designs. Accordingly, different qualities of interaction were found, varying from the lowest quality as chains of individual members’ autonomous actions (Group B, Vignette 1) to strings of synchronized and coordinated actions (Group D, Vignette 2), and finally, contemplative, reflective interactions as a group (Group C, Vignette 3), which could be labeled as the highest quality in terms of interaction in the groups. These different characteristics of the interaction are depicted in Vignettes 1–3. Vignette 1. Group B: chains ofindividual members’ autonomous actions As Table11 shows, in Group B, the second task resulted in a decrease of the average of state changes of social aspects (− 35.2%). The microlevel perspective on interaction data provides a view of a situation comprising chains of localized moments of independence (Davis etal., 2015). Students seemed to have shared intentions (a conceptual goal to solve the problem), but at many moments, they went off individually or acted as varied subgroups to solve the problem (Stahl etal., 2014). While the subgroups worked in parallel, the members communicated their individual efforts. Whereas two of the group members (Lotta and Samuel) focused on task completion individually, Mia did not develop an individual solution, but instead, moved between the other two members, her contributions remaining at a rather superficial level. She mainly asked questions and commented on the work of Lotta and Samuel, without participating in developing their ideas further. Finally, as Samuel failed with his tessellation, Lotta’s work was taken as the final product of their group. In general, the group unsystematically manipulated the tiles to build giant tessellations. The conceptual goals were not communicated, and the mechanical coordination of the group did not seem intentional (Davis etal., 2015). Table 11 Averages of the social and cognitive states in tasks 1 and 2; averages of state changes; and proportional differences of state changes between task 1 and 2 B1 refers to Group B, Task 1, and so forth; “Average” refers to the average of social or cognitive states in Task 1 or 2; “Change” refers to the averages of social or cognitive state changes in Task 1 or 2; and “Difference” points to the proportional difference between the averages of social or cognitive state changes between Tasks 1 and 2 Group/task Social state Cognitive state Average Change Difference (%) Average Change Difference (%) B1 0.115 0.160 0.156 0.117 B2 0.122 0.104 − 35.2 0.171 0.118 0.8 C1 0.135 0.181 0.223 0.101 C2 0.164 0.200 10.7 0.214 0.101 0.1 D1 0.127 0.174 0.246 0.140 D2 0.151 0.173 − 0.3 0.208 0.106 − 24.6 J.Pöysä-Tarhonen et al. 1 3 Vignette 1 Example fromachain ofindividual members’ autonomous activities ingroup interaction Actor Quote 1. Student 1 (Mia) (asking Lotta): “Anything taking shape?” 2. Student 2 (Lotta): “Nothing significant.” 3. Student 1 (Mia) (asking Samuel): “Do you have something, then somehow everything is, like, positioned differently?” 4. Student 3 (Samuel): “I guess so. I don’t know if you can just, like, keep going on with this sort of thing somehow. (Now there’s something starting to –) from there I set out.” 5. Student 1 (Mia) (to Samuel): “So, as I watched that you made –” 6. Student 3 (Samuel): “Now it started to come up here, by this kind of a formula.” [Julia is shaping her pattern; Mia moves to watch Samuel’s working] 7. Student 1 (Mia): “That’s, like, the same (part of the same pattern somehow), isn’t it?” The teacher interrupted the work and reminded the group that they should continue working on the final version of the tessellation. At this stage, Lotta and Samuel continued with their individual tessellations in parallel, communicating over their progression. Actor Quote 8. Student 2 (Lotta): “Well then.” 9. Student 3 (Samuel): “This will be sort of random.” 10. Student 2 (Lotta): “This will be a flame.” 11. Student 3 (Samuel): “Yes, this can be extended.” 12. Student 2 (Lotta): “Help, you’re (Samuel) now coming over to my pattern.” 13. Student 2 (Lotta): “I got inspired to make a flame. I should just, there’s now a partly red and blue flame, then.” 14. Student 3 (Samuel): “[whistling] This is working.” 15. Student 2 (Lotta): “Is it?” 16. Student 3 (Samuel): “Perhaps. Yes, there’s something for all the holes.” Lotta then asks whether the group could merge their individual efforts (mainly her and Samuel’s drafts). Actor Quote 17. Student 2 (Lotta): “Hey, shall we still combine these?” [a short laugh] 18. Student 1 (Mia): “I was thinking that if they should be combined, so that.” 19. Student 3 (Samuel): “A bit difficult, perhaps.” 20. Student 2 (Lotta): “Yes, perhaps.” 21. Student 3 (Samuel): “Because they are at different distances from each other. We would need to move the whole pattern. It wouldn’t really be feasible.” “Anything taking shape?” Capturing various layers ofsmall… 1 3 Actor Quote 22. Student 1 (Mia) (to Lotta): “Take some yellow from there now. (Would it be) still like the same pattern, with different colors? As there’s a blue center –” 23. Student 3 (Samuel): “Hey, now there’s a problem. Now it’s not working anymore.” Vignette 2. Group D: strings ofsynchronized andcoordinated actions Table11 shows how, in Group D, the second task resulted in a decrease in the average of cognitive states and the average of state changes (− 24.6%). When zooming back in on the group interaction, it seems that the group stayed together over the process and jointly discussed and decided what they were doing or should do; however, the goals were less targeted and immediate. The group members constantly externalized their thoughts, which was not the case in Group B. At the beginning, the group discussed their previous session and decided to set more goals and work more logically than previously. Here, they discovered a “diamond problem,” as they called it; that is, they found themselves in a situation in which they would have needed extra, diamond-shaped tiles to fill all the gaps in their designed patterns (see also Hähkiöniemi etal., 2016). Vignette 2 Example fromsynchronized andcoordinated actions ofthegroup (Group D) Actor Quote 1. Student 1 (Leo): “((laughs)) Then we would only need a sort of small diamond shape so that we could in a way make this work.” 2. Student 4 (Anna): “((laughing)) Yes.” 3. Student 2 (Lisa): “So, this isn’t really like working in any way now, is it?” 4. Student 1 (Leo): “Nope.” 5. Student 4 (Anna): “No, it isn’t.” 6. Student 1 (Leo): “No, as it calls for a sort of diamond. Then it would become like that.” 7. Student 3 (Nea): “Diamond-shaped sweets.” 8. Student 1 (Leo): “Or, yes, diamond shapes for here, so then it would become like that.” 9. Student 4 (Anna): “Yes.” 10. Student 3 (Nea): “Well what if, could that one be set somehow in another direction, or like, could we get it here somehow?” 11. Student 2 (Lisa): “Yes.” 12. Student 4 (Anna): “But isn’t that-” 13. Student 1 (Leo): “That’s quite an impossible angle; you cannot really do anything about it. ((Notices Nea’s solution)) No, I mean, you actually can do it that way.” 14. Student 4 (Anna): “Yes, you actually can.” 15. Student 2 (Lisa): “But I don’t know, I think that the problem will always come up at some point, the problem that -” J.Pöysä-Tarhonen et al. 1 3 Actor Quote 16. Student 4 (Anna): “That it requires some supplementary job there.” 17. Student 3 (Nea): “The diamond.” Vignette 3. Group C: contemplative, reflective interactions asagroup As Table 11 shows, in Group C, the second task resulted in an increase in the average of social states and a slight increase in the social state changes (10.7%). Accordingly, the microlevel perspective on the interaction data of Group C provides us with a view of contemplative interactions in which the group members vocalized both immediate conceptual goals and goals going beyond the moment-to-moment interactions (Davis etal., 2015). The overarching goals for their discoveries were verbalized explicitly by relying on mathematical concepts, and the group was willing to verify the final pattern with GeoGebra. As a “warm-up” exercise, the group started by forming a pattern similar to the one made in the previous session with GeoGebra. Next, the group created a new, star-shaped pattern. The students used mathematical terms (i.e., reflection, rotation, translation), but they also wanted to make the pattern esthetic via the use of colors. To ensure the final pattern was mathematically correct, the group decided to test it with GeoGebra. The next excerpt is from the phase after the warm-up exercise, when the group started working on the real pattern. The group was oriented toward joint problem solving and communication, with individual members building on the words and ideas of their co-members (Stahl etal., 2014). The central, primary idea seems to have originated from Student 1 (Linus). Vignette 3 Example fromContemplative, Reflective Interactions intheGroup (Group C) Actor Quote 1. Student 1 (Linus): “I was thinking that, if you make a straight line between these points, you know. No, but it won’t work because (–).” 2. Student 3 (Tina): “So, is it like, so that—” 3. Student 1 (Linus): “The way it goes, yes. 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Authors and Affiliations JohannaPöysä‑Tarhonen1 · PäiviHäkkinen1 · PasiTarhonen2· PiiaNäykki3,4 · SannaJärvelä4 Päivi Häkkinen [email protected] Pasi Tarhonen [email protected] Piia Näykki piia.t.na[email protected] Sanna Järvelä [email protected] 1 Finnish Institute forEducational Research, University ofJyvaskyla, P.O. Box35, 40014Jyvaskyla, Finland J.Pöysä-Tarhonen et al. 1 3 2 Honeywell Inc., P.O. Box1001, Viestikatu 1-3, 70601Kuopio, Finland 3 Faculty ofEducation andPsychology, University ofJyväskylä, P.O. Box35, 40014Jyvaskyla, Finland 4 Faculty ofEducation, University ofOulu, P.O. Box8000, 90014Oulu, Finland