The interplay between the guidance from the digital learning environment and the teacher in supporting folding back

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/ The interplay between the guidance from the digital learning environment and the teacher in supporting folding back © The Author(s) 2022 Published version Hähkiöniemi, Markus; Francisco, John; Lehtinen, Antti; Nieminen, Pasi; Pehkonen, Salla Hähkiöniemi, M., Francisco, J., Lehtinen, A., Nieminen, P., & Pehkonen, S. (2023). The interplay between the guidance from the digital learning environment and the teacher in supporting folding back. Educational Studies in Mathematics, 112(3), 461-479. https://doi.org/10.1007/s10649-022-10193-x 2023 Vol.:(0123456789) Educational Studies in Mathematics https://doi.org/10.1007/s10649-022-10193-x 1 3 The interplay betweentheguidance fromthedigital learning environment andtheteacher insupporting folding back MarkusHähkiöniemi1 · JohnFrancisco2 · AnttiLehtinen1,3 · PasiNieminen1 · SallaPehkonen1 Accepted: 3 October 2022 © The Author(s) 2022 Abstract Previous studies have proposed that students’ mathematical understanding develops dynamically through the process known as folding back, in which learners revisit earlier forms of understanding and use them to build even deeper levels of mathematical understanding. Digital learning environments, where students can manipulate representations, are often used to enable students to notice properties, patterns, or rules. When working in such an environment, students usually receive support from the environment and the teacher. The interplay between these different sources of support is important according to previous studies. In this study, we examine this interplay in the case of folding back. The study aims to understand how the teacher, together with the learning environment, can support the process of folding back. We collected data from second, fourth, and sixth grade students as they worked in groups to develop a rule for balancing a balance beam in a digital learning environment designed to support folding back. One pre-service teacher guided each threestudent group. Data were analyzed by identifying occasions for folding back and characterizing different ways in which the interplay between the teacher and the environment supported students’ folding back. We found different kinds of synergy between the two sources of support. The teachers followed up on and augmented the support from the environment, initiated supplementary folding back, and reinforced the support from the environment. We also found non-synergy between the two sources of support, when the teachers’ support was not aligned with support from the environment. Keywords Folding back· Distributed support· Digital learning environment· Synergistic support· Teacher support· Technology * Markus Hähkiöniemi [email protected] 1 Department ofTeacher Education, University ofJyvaskyla, Jyväskylä, Finland 2 College ofEducation, University ofMassachusetts, Amherst, MA, USA 3 Department ofPhysics, University ofJyvaskyla, Jyväskylä, Finland M.Hähkiöniemi et al. 1 3 1 Introduction The use of digital learning environments is common practice in schools. In mathematics, digital environments help students notice connections, properties, patterns, or rules as a result of manipulating representations (e.g., Erbas & Yenmez, 2011; Olsson & Granberg, 2019). Despite working in digital environments, students still need teachers to support their mathematical reasoning (Drijvers etal., 2010; Hähkiöniemi etal., 2013). A key process in mathematical understanding is the concept of folding back (Martin, 2008; Pirie & Kieren, 1994). When students face a challenge while working on a mathematical task, they may need to fold back to revisit earlier understandings in ordertothen build even deeper understandings and use them to overcome the challenge. Except for some research initiatives (e.g., Martin, 2008; Martin & Towers, 2016a; Yao & Manouchehri, 2022), few studies have examined the role of the teacher or technology in supporting folding back. Research on distributed scaffolding has shown that student learning may be supported by multiple sources, such as a teacher and learning materials (Brown etal., 1993; Kolodner etal., 2003). This line of research has pointed out that considering the interplay between different sources of support is important (Martin etal., 2019; Tabak, 2004). According to Tabak (2004), synergy between sources can contribute to productive support. Teachers or other social supports play an important role in making sure that the support is responsive to students’ needs by complementing the support from the environments with their own support (Puntambekar, 2022). Consistent with this position, Martin etal. (2019) call for more research on understanding the interplay between support from a teacher and material sources. Drawing on insights from the research on distributed scaffolding, more attention could be given to interplay between different sources of support in thecase of folding back. So far, no studies have investigated the interplay between sources of support in guiding students’ folding back. Yet, this kind of research is needed to advance our understanding of how to support mathematical understanding through folding back and to improve the design of learning environments. This study addresses this research gap by examining the interplay between support from the teacher and from a digital learning environment in promoting primary school students’ folding back when the students explore a rule for balancing a balance beam. The following research question guided the study: What relations exist between guidance from the teacher and the digital learning environment in cases of supporting folding back? 2 Theoretical background 2.1 Promoting mathematical understanding andfolding back The Pirie-Kieren theory of growth of mathematical understanding is a well-established theoretical perspective on the nature of mathematical understanding (see, e.g., Kieren etal., 1999; Martin & Pirie, 2003; Pirie & Kieren, 1994). According to this theory, growth of mathematical understanding is neither linear nor monodirectional. Instead, it is the result of a dynamical and active process involving a continuous movement back and forth between eight layers of understanding (see Fig.1). The four layers that apply to the present study are summarized in Fig.1. The interplay betweentheguidance fromthedigital learning… 1 3 In the Pirie-Kieren theory, when faced with a problem that is not immediately solvable, a person functioning at an outer layer of understanding may need to revisit an inner layer of understanding to examine and modify their current ideas and thinking about a concept. This process is known as “folding back” to imply that when learners revisit an earlier layer of understanding, they carry with them understandings from the outer layer, a phenomenon called “thickening” (Fig.1). Two key features define folding back and differentiate it from a simple act of going back: The return to an inner layer is “stimulated and guided by outer level knowing” and the”folding back allows for the reconstruction and elaboration of inner level understanding to support and lead to new outer level understanding” (Pirie & Kieren, 1991, p. 172). Folding back promotes understandings because when learners revisit an earlier understanding in response to a challenge, they can modify, collect, or build anew conceptions that will allow the difficulty to be overcome through an extended understanding of the topic (Martin, 2008). 2.2 Supporting folding back Martin (2008) developed a framework for folding back intended as an observational analytical tool. The framework identifies three higher-level categories that describe key aspects of folding back. Source refers to the stimulus that prompts the learner to fold back and four sources are identified: the teacher (teacher intervention), another student (peer intervention), curriculum material (material intervention) or the student who decides to fold back (self-invoked). Form refers to the kinds of actions engaged at the inner layer (e.g., collecting an existing understanding). Outcome refers to the effect of folding back on growth of students’ understanding (e.g., effective or not in enabling continued growth). Each category is subdivided into subcategories to provide further descriptions of folding back. Based on Martin’s framework, support for folding back consists of creating occasions for and stimulating folding back, helping learners engage in appropriate inner layer activity, and making sure that learners return to the outer layer and use the extended understanding to solve the challenge that motivated the folding back. In this study, we focus on two support sources: the teacher and the material. Fig. 1 Layers of understanding and an instance of folding back in a hypothetical path of growth of understanding (Pirie & Kieren, 1994) M.Hähkiöniemi et al. 1 3 Research on teacher support for folding back shows that teachers can support folding back by using moves such as rug-pulling, which shifts the focus of students’ attention to something that confuses them and requires them to reassess their mathematical process (Towers, 1998; Towers & Proulx, 2013), and focusing students’ attention on contradictions in their current understanding (Hähkiöniemi & Hirvonen, 2013). Martin and Towers (2016a) reported a case of a high school teacher who encouraged folding back by getting students to revisit and build on ideas from previous studies. Teachers can also use folding back as a pedagogical design tool for planning theirteaching to create occasions for folding back (Martin & Towers, 2016a, b). Research on support for folding back from material sources are rare. However, Gulkilik etal. (2020) showed that virtual manipulatives can support a student’s understanding as defined in Pirie-Kieren theory. Poon and Wong (2017) designed dynamic geometry materials that encouraged students to fold back. They found that materials provided learning opportunities. Yao and Manouchehri (2022) found that technology may mediate folding back initiated by the teacher or by the students themselves. However, studies have not elaborated on the interplay between teachers and material in supporting folding back. 2.3 Synergy betweendifferent sources ofsupport Research on distributed scaffolding has contributed to understanding the interplay between sources of support. Tabak (2004) described two patterns of distributed scaffolds that existed in the research literature. In the differentiated scaffolds pattern, support is provided through different means to address diverse learning needs. In the redundant scaffolds pattern, different sources support the same learning need at different points in time. Tabak introduced a third pattern, the synergistic scaffolds, in which multiple sources of support interact to target the same learning need at the same time. Tabak argues that it is an “important conceptual tool in understanding how different constituents interact to produce support that is greater than the sum of the constituents” and “the central question is not whether interaction between supports can occur, but how this interaction can come into play and what functions it can serve” (Tabak, 2004, p. 308). While Tabak emphasized the importance of synergistic scaffolds, he noted that it has not received much attention in research. In mathematics education, the synergy between sources of support has been examined only in some studies (Tropper etal., 2015; van Zoest & Stockero, 2008). When sources of support include the teacher, the teacher may be particularly important in interpreting the situation with respect to students’ thinking and use of the other sources of support. Puntambekar etal. (2007) highlighted the teacher’s role in building connections between different materials and activities. As an example of the teacher’s role, Lehtinen and Hähkiöniemi (2016) found that the use of technology may create occasions for productive student explanation, but how this opportunity is exploited depends on teachers’ complementary guidance. Lehtinen and Viiri (2017) provide an example of synergistic guidance in which pre-service teachers advised students on selecting an appropriate difficulty level in a game. Furthermore, findings by Martin etal. (2019) point to the importance of complementarity between support from the teacher and material sources. Teacher support that only replicated guidance provided by the software was not as productive as teacher support that provided augmenting guidance (Martin etal., 2019). The interplay betweentheguidance fromthedigital learning… 1 3 3 Methods 3.1 Design ofthelearning environment We developed a digital learning environment by using the Graasp authoring platform (Graasp, 2021) and GeoGebra (GeoGebra, 2021). The environment consists of seven tabs (Table1) in which weights and locations of two birds on a balance beam can be changed while the beam is supported to stay in balance. When the supports are removed, the beam stays in balance or tilts. In Lab (Fig.2), students can experiment freely and start building an image about the functioning of the balance beam,by noticing properties in the balance cases and formulating rules. It was expected that at first, students would develop lowerlevel rules such as weights and distances being equal. In the tasks, students are asked to apply their rule (shown above the task in noneditable form). If they solve a task correctly so that the beam stays in balance, the environment prompts them to move on to the next task. If they do not succeed, the screen turns into gray, the birds cannot be moved, and they receive a prompt to return to Lab to develop their rule before returning to the task. To avoid using trial and error, Fig. 2 Lab M.Hähkiöniemi et al. 1 3 Table 1 Weights and distances of the birds on a balance beam in different tabs Bird on the left Bird on the right Assignment Weight Distance Weight Distance Intro Fixed, not shown Not shown Fixed, not shown Not shown Try how to move the birds and how to remove the supports Lab 1–6kg 1–8m 1–6kg 1–8m Formulate a rule to balance the seesaw with. Experiment with multiple weights and locations for the birds Task 1 7kg 2m 1–20kg 1–8m Use your rule to balance the beam. You have only one try Task 2 12kg 1m 1–7kg 1–8m Use your rule to balance the beam. You have only one try Task 3 3kg 2–8m 9kg 2–8m Use your rule to balance the beam. You have only one try Task 4 6kg 1–8m 9kg 4m Use your rule to balance the beam. You have only one try Additional tasks Balancing Act PhET simulation (PhET Interactive Simulations, 2021) E.g., balance the beam, decide whether the beam tilts, find the weight of an object The interplay betweentheguidance fromthedigital learning… 1 3 the number of tries in the tasks was limited by locking the screen after an answer. To enable opening the task again after visiting in Lab, a start-button appears on the corner of the screen after 7s. The tasks were designed to require progressively more advanced rules. For example, task 2 can be solved using a rule that involves theproportion of weights being 1:2, while the proportion of weights is 1:3 in task 3 and 2:3 in task 4. Thus, it was expected that at some point students would solve a task incorrectly, which would point to a limitation in their current rule at Property Noticing (rule is tied to particular cases) or at Formalising (general rule). It was assumed that students would fold back to Image Making by experimenting in Lab and finally improve their rule so that the task could be solved. Thus, the sequence of the tasks, the possibility to experiment in Lab and the prompt to return to Lab were expected to support folding back. 3.2 Data collection The students were randomly divided into groups of three students per laptop. Two groups included second grade (8-year-old) students, five groups included fourth grade (10-year- old) students, and five groups included sixth grade (12-year-old) students. They worked in these groups for the full duration of the lesson (approximately 40min). Each group had one pre-service primary school teacher guiding their work. Thus, 12 pre-service teachers participated in the study. The 12 pre-service teachers were participants in a course that focused on guiding students based on their thinking when using dynamic representation for learning mathematics and science. One session (2.5h) was devoted to familiarizing with the balance beam activity and preparing for guiding students. The pre-service teachers used the same environment as students. Ideas about different rules and what would prompt someone to modify a rule were collected. The participants also discussed how to interpret certain hypothetical rules and what kind of support could be provided. The concept of folding back was not discussed, but, the idea of building rules based on empirical experimenting in Laband trying to apply these rules in the tasks, then returning to experiment in Lab when the rule did not help to solve a task, were discussed. The screen of each laptop was recorded using screen capture software. The software also captured audio from the laptop microphone and video from the laptop webcam in sync with screen capture. In addition, a small action video camera recorded the group from the side to enable the recognition of gestures and who was talking. All these data sources from each group were synchronized in one video file when preparing the data for analysis. 3.3 Data analysis We used the video analysis model developed by Powell etal. (2003) as the model suits the purpose of developing insights and building an understanding of complex phenomena. The heart of this method is selecting critical events that help build insights related to the research question—in this case, the interplay between the teacher and the learning environment in supporting folding back. M.Hähkiöniemi et al. 1 3 In this study, the critical events were episodes in which the teacher or the environment pointed to a limitation in students’ current rules and prompted the students to experiment more. In these events there was an opportunity for folding back to Image Making, although it did not always happen. Image Making was interpreted when students experimented with the balance beam (whether in Lab or in tasks). When working in the environment, students can fold back to various layers, but the analysis focused particularly on folding back to Image Making because the environment was designed to support folding back primarily to that layer. The analysis consisted of six steps. First, data were transcribed and imported into video analysis software together with the video for each student group. Second, we familiarized ourselves with the data by viewing videos in parallel with transcripts. Third, we described the students’ development of ideas and teacher guidance to achieve an overall picture of each group. Fourth, we identified critical events in which there was an opportunity for folding back to Image Making. Fifth, we analyzed the critical events for how the teacher and the environment supported (or not) folding back. In connection with this, we examined how the teacher complemented support from the environment. Sixth, we recognized and characterized different ways in which the interplay between the teacher and the environment supported the students’ folding back to Image Making. 4 Results We found that the interplay between the teacher and the learning environment in supporting folding back appeared in three synergistic and two non-synergistic ways. 4.1 Synergistic guidance In synergistic guidance, the teachers’ support complemented the support from the environment so that the two sources of support worked in concert. Both sources shared the same aim of sending students from tasks back to Lab to continue image making when needed. 4.1.1 The teacher follows uponandaugments support fromtheenvironment Often, students’ progress in the learning environment created an occasion where students needed to revise their rule for balance and the environment prompted them to fold back, but, the students did not follow the prompt. However, the teachers followed up on the prompt from the environment, added support that was adjusted to students’ work, and guided the students to fold back as intended by the environment. For example, Evan, Lucy, and Landon (sixth grade) built their image by experimenting in Lab and noticed the following property: “the heavier bird is closer and the lighter bird is further away from the center.” Related to this, Lucy added that “If it is 2kg heavier, it has to be two (inaudible: toward there).” They solved task 2 incorrectly (12kg 1m ∧ 7kg 5m). Despite the environment prompting the students to return to Lab, they continued to think about task 2: 1 Lucy: Or if it has to be 5 between them, because 12 – 7 = 5. Maybe there has to be 5 between them, like 1, 2, 3, 4, 5 [points to 5 jumps from the bird on the left and ends at 4 m]. The interplay betweentheguidance fromthedigital learning… 1 3 23 Mr. Lopez: What is 9 × 4? 24 Dylan: 36. 25 Mr. Lopez: Yeah. 26 Dylan: Um, they are the same. Based on their image and the teacher’s hint (turn 1), Benjamin suggested a particular multiplication to have something to do with the balance (turn 4). The students were at Image Having as they had some idea about how the balance beam works and stated only a partial relation between the variables. The teacher asked the students to continue this idea (turns 6–14). When Dylan (turn 15) and Benjamin (turn 18) gave some suggestions, the teacher’s feedback suggested tothem that they think about something else (turns 16 and 19). Finally, the teacher suggested calculating particular multiplications (turns 19 and 21) and the students noticed the products to be equal. Thus, the teacher guided the students to use the teacher as the source for finding the rule. It may seem as if the students’ folded back to Image Making when they suggested something and the teacher gave feedback. However, the students did not work on their images by experimenting. Rather, they gave unconnected suggestions based on the teacher’s hints until they reached the point where the teacher was aiming at. Thus, students did not fold back to Image Making. 5 Discussion We set out to explore the interplay between guidance provided by theteacher and guidance provided by a digital learning environment in supporting folding back. The students’ interactions with the digital environment created instances of folding back as intended in the design of the environment. Thus, our results support suggestions that material sources can be designed to intentionally launch folding back (Martin, 2008; Poon & Wong, 2017; Yao & Manouchehri, 2022). In our study, material support made it possible that the need for folding back emerged from the students’ work in the environment. However, teachers had to complement material support in several ways, which was crucial for folding back. The teachers were able to adapt their support to how the students received the prompts from the environment and how the students worked. Thus, this study supports the importance of teachers and digital learning environments supporting students in a synergistic way (Martin etal., 2019; Tabak, 2004; Tropper etal., 2015). Furthermore, in the case of synergistic support for complex and challenging processes, such as folding back, our results highlight the importance of teachers adapting support for students. In this study, material support alone seemed insufficient for creating folding back. The reason may be that initiating folding back involves pointing to limitations in students’ current understanding and prompting them to take a step back. Productivity of this kind of move, that makes students’ work more problematic, may depend on further guidance (Reiser, 2004). Indeed, our results show several ways in which further adaptive guidance from the teacher was important for supporting folding back. We identified three ways in which the teacher and the environment can synergistically support folding back. First, the environment prompted folding back, and the teacher followed up with adaptive guidance. The prompt from the environment was based on the students’ work, and the students saw that their work had some drawbacks. However, for the folding back to actually happen, it was important that the teachers provide further support by taking into account the students’ thinking and how they reacted to the prompt. This synergy helped the students react to the prompt from the environment in a productive manner. M.Hähkiöniemi et al. 1 3 This is similar to how Tabak (2004) described that teacher modeling augmented software tools so that culturally appropriate uses of the tools were made visible to the students. Second, the teachers initiated supplementary instances of folding back on the fly. Here, the teachers augmented the support for folding back, as the support preplanned in the tasks and theprompts from the environment were not enough. Unlike the environment, the teachers could interpret students’ thinking behind a correct answer and create an occasion for folding back when needed. Third, the teachers reinforced the guidance from the environment by repeating the prompts from the environment. Even with this kind of simple guidance, the teachers augmented the support of the environment by insisting that students actually engage in image making and not just quickly visit Lab. This augmentation does not require a thorough analysis of students’ ideas, but it still shows teachers adapting their support to students’ general ways of receiving the prompt. Although Martin etal. (2019) found that replicating the guidance provided by the software was not as productive as complementing the guidance, we found it important that the teachers insist that students follow the guidance from the environment. This kind of reinforcing may be particularly important in supporting folding back, as the purpose is to challenge the students as opposed to offering help. We also found instances of non-synergistic guidance where the teachers’ support was not aligned with support from the environment. First, the teachers contradicted the prompt from the environment intended to support folding back to Image Making in Lab and guided the students to solve tasks through experimenting. This led the students to fold back to Image Making within the task. Instead of open exploration in Lab, the teachers supported searching for one balance case within the constraints of the task. This is similar to Tower’s (1998, 2002) blocking move, where the teacher blocked the potential folding back that was about to happen. Towers questioned the productivity of this move. However, in our study the move seemed to be productive when the teacher blocked a particular kind of folding back prompted by the environment andinstead supported another one. Yet, we also found that experimenting within a task can change the nature of the activity so that students only solve the tasks through trial and error without the aim of developing a rule. This is similar to what Martin (2008) calls “going back,” as the students go back to Image Making, but they do not connect their new image making with their previous understanding at the outer layer, and thus, their understanding is not thickened. Second, instead of supporting folding back to Image Making by experimenting, the teachers started to give hints for the rule. This led the students to focus on guessing what the teacher was thinking instead of exploring the balance themselves. It seemed as if the students were using the teacher as an experimenting device. Through this process the students expressed a sophisticated rule, but this was not based on image making. Instead, the students just adopted the new rule that was not connected to their understanding at the inner layers. Pirie and Kieren (1994) warned that, if a teacher offers information in a ready-made form, it may lead to adisjointed piece of understanding that may be difficult to use in building further understanding. While this study shows that material support can be designed to intentionally support folding back, it is important that teachers share this intention. Non-synergistic guidance shows that teachers may change the nature of the activity. Thus, our results support Tabak’s (2004) suggestion that productive synergy requires teachers’ conceptions to be consistent with material support. However, our results also show that even non-synergistic guidance where ateacher contradicts the guidance from the environment can productively support folding back. As an implication for practice, the results suggest that folding back can be used as a design principle for creating digital learning environments in asimilar wayto how Martin The interplay betweentheguidance fromthedigital learning… 1 3 and Towers (2016a) described the use of folding back as a pedagogical tool in planning lessons. Environments can be planned not only to help students, but also to purposefully make things more problematic for students at an appropriate point in the growth of their understanding. However, in the case of folding back, the teacher plays an important role in complementing support from the environment. Thus, learning environments can be enriched by planning synergistic guidance from the environment and the teacher. Teachers could be prepared to follow up on and augment the guidance from the environment, launch supplementary folding back, and reinforce the prompts from the environment. In addition, suggestions for improving the design of this and similar digital learning environments can be drawn from the study. To emphasize theimage making activity, the environment could automatically transfer the students to Lab after an incorrect answer to a task. To emphasize solving tasks by applying the rule, students could be prompted to explain how the rule is used and to return to Lab even before answering the task if they cannot use their current rule. However, despite any improvements that can be made, teachers will still play an important role in supporting understanding in digital environments, as they can establish synergistic interactions with the environment that ensure that support is responsive to students’ needs. Funding Open Access funding provided by University of Jyväskylä (JYU). This work was funded by the Academy of Finland (project number 318010). Data availability statement The data is not publicly available. Declarations Conflict of interest The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. References Brown, A. L., Ash, D., Rutherford, M., Nakagawa, K., Gordon, A., & Campione, J. C. (1993). 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