Education, Faculty of
Curriculum and Pedagogy (EDCP), Department of
DSpace
UBCV
Zhang, Ting
Doctor of Philosophy - PhD
University of British Columbia
In this case study, I investigated young children’s learning experiences with robotics in after-school learning centers. Over a two-month period, six students in grades 2 and 3, currently enrolled in these learning centers, constituted four subcases (two pairs and two singles) in this study, with each participating in six one-hour robotics sessions. The study applies the principles of constructionism and self-regulated learning theory to address two primary questions: i) How did these young children apply mathematical concepts while engaging in robotics activities? ii) What self-regulated learning opportunities does robotics provide for young children during problem-solving activities? The findings revealed that students exhibited an iterative approach in their problem-solving endeavors, involving continuous cycles of understanding the problem, ideation, experimentation, and assessing solutions. While coding and designing robots, they employed strategies such as decomposition, experimentation, and self-assessment to gauge their progress. Moreover, they showcased their ability in applying mathematical concepts across various representations, surpassing mere basic arithmetic computation. Additionally, the study found that the children as young as six years old consistently self-regulated their problem-solving, encompassing crucial components of SRL such as goal-setting, self-assessment, and persistence in the face of challenges. These insights contribute to our understanding of how students engage in problem-solving within the context of robotics and hold implications for future research and educational practices in mathematics education.
EXPLORING CHILDREN’S MATHEMATICS LEARNING AND SELF-REGULATION IN ROBOTICS by Ting Zhang A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Curriculum Studies) THE UNIVERSITY OF BRITISH OF COLUMBIA (Vancouver) April 2024 © Ting Zhang, 2024 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Exploring Children’s Mathematics Learning and Self-Regulation in Robotics submitted by Ting Zhang in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Curriculum Studies Examining Committee: Dr. Ann Anderson, Professor, Department of Curriculum and Pedagogy, UBC Supervisor Dr. Cynthia Nicol, Professor, Department of Curriculum and Pedagogy, UBC Supervisory Committee Member Dr. Deborah Butler, Professor, Department of Educational & Counselling Psychology, and Special Education, UBC Supervisory Committee Member Dr. Nancy Perry, Professor, Department of Educational & Counselling Psychology, and Special Education, UBC University Examiner Dr. Susan Gerofsky, Associate Professor, Department of Curriculum and Pedagogy, UBC University Examiner Dr. Viktor Freiman, Professor, Faculty of Educational Sciences, University of Moncton External Examiner iii Abstract In this case study, I investigated young children’s learning experiences with robotics in after-school learning centers. Over a two-month period, six students in grades 2 and 3, currently enrolled in these learning centers, constituted four subcases (two pairs and two singles) in this study, with each participating in six one-hour robotics sessions. The study applies the principles of constructionism and self-regulated learning theory to address two primary questions: i) How did these young children apply mathematical concepts while engaging in robotics activities? ii) What self-regulated learning opportunities does robotics provide for young children during problem-solving activities? The findings revealed that students exhibited an iterative approach in their problem-solving endeavors, involving continuous cycles of understanding the problem, ideation, experimentation, and assessing solutions. While coding and designing robots, they employed strategies such as decomposition, experimentation, and self-assessment to gauge their progress. Moreover, they showcased their ability in applying mathematical concepts across various representations, surpassing mere basic arithmetic computation. Additionally, the study found that the children as young as six years old consistently self-regulated their problem-solving, encompassing crucial components of SRL such as goal-setting, self-assessment, and persistence in the face of challenges. These insights contribute to our understanding of how students engage in problem-solving within the context of robotics and hold implications for future research and educational practices in mathematics education. iv Lay Summary In this study, I investigated the experiences of elementary students in Grades 2 and 3 who participated in robotics activities at after-school learning centers. I looked at how children used math while engaging with robotics and what self-regulated learning opportunities these robotics activities offered. The results showed that students went through a dynamic iterative problem-solving process which was not step-by-step linear. They used strategies like breaking it down, trying it out, and self-evaluating their progress. The combination of physical construction and coding in robotics activities also allowed them to interact with various math concepts in a multidimensional manner. I observed that students self-regulated their learning by setting goals, taking strategic actions, and being persistent in the face of failures. These findings provide insights into how students integrated mathematics and self-regulation in after-school robotics activities and have implications for future research and educational practices in math education. v Preface The study reported in this dissertation was designed, conducted, and analyzed by the author with guidance from the supervisory committee. No part of this research has been published. This research study has been approved by the University of British Columbia Behavioral Ethics Board (Certificate Number: H18-02826). vi Table of Contents Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ......................................................................................................................... vi List of Tables ..................................................................................................................................x List of Figures ............................................................................................................................... xi Acknowledgments ...................................................................................................................... xiii Dedication .....................................................................................................................................xv Chapter 1: Introduction ................................................................................................................1 1.1 Statement of Problem ....................................................................................................... 1 1.2 Research Questions .......................................................................................................... 3 1.3 Significance of the Study ................................................................................................. 4 Chapter 2: Literature Review .......................................................................................................6 2.1 Theoretical Framework .................................................................................................... 7 2.2 Mathematical Problem Solving ...................................................................................... 13 2.2.1 Polya’s Problem Solving Model ........................................................................... 16 2.2.2 Self-Regulation Models ........................................................................................ 19 2.2.3 Computational Thinking ....................................................................................... 22 2.2.4 Engineering Design Process ................................................................................. 25 2.3 Technologies Integration in Mathematics Education ..................................................... 27 vii 2.4 Educational Robotics in Mathematics Education........................................................... 29 Chapter 3: Methodology..............................................................................................................33 3.1 Settings ........................................................................................................................... 35 3.2 Participants ..................................................................................................................... 38 3.3 Robotics Activities ......................................................................................................... 41 3.4 Researcher’s Role ........................................................................................................... 48 3.5 Data Collection ............................................................................................................... 50 3.5.1 Observation ........................................................................................................... 50 3.5.2 Interviews: Problem-solving, Video Recall .......................................................... 51 3.5.3 Artifacts................................................................................................................. 53 3.5.4 Data Analysis ........................................................................................................ 54 Chapter 4: Results........................................................................................................................58 4.1 Iterative Problem-Solving in Robotics ........................................................................... 58 4.1.1 Individual Problem Solver in Action .................................................................... 59 4.1.2 Cross-Case Problem Solving Approach: The Iterative Process ............................ 68 4.1.3 Strategies for Problem-Solving in Robotics ......................................................... 76 4.2 Choice Making and Goal-setting in Robotics ................................................................ 87 4.3 Applications and Perceptions of Mathematics in Robotics ............................................ 92 4.3.1 Students’ Applications of Mathematics in Robotics ............................................. 92 4.3.2 Students’ Perception of Mathematics in Robotics. ............................................... 98 viii 4.4 Recognition and Perceptions of Errors in Problem-Solving ........................................ 100 4.4.1 Students’ Recognition of Errors while Building and Coding Robots ............. 100 4.4.2 Students’ Responses to Errors ........................................................................ 103 4.5 Chapter Summary ......................................................................................................... 107 Chapter 5: Discussion ................................................................................................................110 5.1 Children’ Use of Mathematics in After-school Robotics ............................................. 110 5.2 Children’s Self-Regulated Learning in Robotics ......................................................... 114 5.3 Implications for Practice .............................................................................................. 117 5.3.1 Incorporating Non-Routine Tasks in Robotics Activities ................................... 118 5.3.2 Embracing Chaos in Robotics Problem-Solving ................................................ 118 5.3.3 Supporting SRL in Robotics in After-School Programs ..................................... 120 5.3.4 Connecting Robotics to Mathematics in School Classrooms ............................. 121 5.4 Implications for Future Research ................................................................................. 122 5.5 Conclusion .................................................................................................................... 124 References ...................................................................................................................................126 Appendices ..................................................................................................................................146 Appendix A: Sample Problem Solving Activities Screenshots .............................................. 146 A.1 Windmill (clock option) Screenshot ............................................................................ 146 A.2 Butterfly Project Screenshot ........................................................................................ 146 ix Appendix C: Field Notes ........................................................................................................ 148 Appendix D: Interview Questions .......................................................................................... 149 Appendix E: Interview Excerpt .............................................................................................. 150 x List of Tables Table 3.1 Summary of Participant Information ......................................................................... 40 Table 3.2 Activities and Sites .................................................................................................... 45 Table 4.1 Problem-Solving Phases and Examples ..................................................................... 69 Table 4.2 Problem-Solving Iteration Example .......................................................................... 73 xi List of Figures Figure 2.1 Illustration of Polya’s 4-step Problem-Solving Model ............................................. 17 Figure 2.2 An Integrative Model of Self-Regulation ................................................................. 21 Figure 2.3 Engineering Design Process Diagram ...................................................................... 25 Figure 3.1 A Classroom and Workstations ................................................................................ 36 Figure 3.2 Samples of Special Building Elements..................................................................... 37 Figure 3.3a An Example Page of Robotics Curriculum ............................................................ 41 Figure 3.3b Screenshots of two buttons associated with the Butterfly icon .............................. 42 Figure 3.4 Area for Challenge Stage and Interviewing ............................................................. 44 Figure 3.5 A Screenshot of the Programming Block Palette ..................................................... 53 Figure 3.6 A Sample of Coding Script....................................................................................... 54 Figure 4.1.1 The Assembly Windmill Base .............................................................................. 60 Figure 4.1.2 Sylvia’s Clock ....................................................................................................... 61 Figure 4.1.3 Development of Sylvia’s Clock Code .................................................................. 63 Figure 4.1.4a Tania’s Clock Face .............................................................................................. 64 Figure 4.1.4b Tania’s Clock ....................................................................................................... 64 Figure 4.1.4c Tania’s Clock Code ............................................................................................. 65 Figure 4.1.5 Carl and Princeton’s Windmill and Coding Script ................................................ 66 Figure 4.1.6 Jason and Kent’s Windmill and Coding Script ..................................................... 68 Figure 4.1.7a Iconic Representation of Understanding, Ideation, Experimentation, & Assessment Cycle .......................................................................................................................................... 71 Figure 4.1.7b Each Subcase’s Iterative Problem Solving During Challenge in Windmill/Clock.................................................................................................................................................... 72 xii Figure 4.1.8 See-saw/Hammer Excerpt from Table 3.2 ............................................................ 76 Figure 4.1.9 Sample of Sylvia’s Decomposition ....................................................................... 77 Figure 4.1.10 Frog Excerpt from Table 3.2 ............................................................................... 78 Figure 4.1.11 Sample of Tania’s Decomposition ...................................................................... 78 Figure 4.1.12 Bulldozer Excerpt from Table 3.2 ....................................................................... 79 Figure 4.1.13a Carousel excerpt from Table 3.2 ....................................................................... 80 Figure 4.1.13b Trial and Error: Jason and Kent’s Carousel Coding Script ............................... 80 Figure 4.1.14 Windmill Excerpt from Table 3.2 ....................................................................... 81 Figure 4.1.15 Cross-Checking Calculation ................................................................................ 84 Figure 4.2.1 Tania’s Walker Code ............................................................................................. 88 Figure 4.3.1 Sylvia’s Robotics Butterfly ................................................................................... 93 Figure 4.3.2 Sylvia’s Butterfly Code ......................................................................................... 94 Figure 4.3.3 Clock Faces Made by Students (Left: Sylvia, Right: Tania) ................................. 95 Figure 4.3.4 Sylvia’s Clock code ............................................................................................... 96 Figure 4.3.5 Tania’s Clock Code ............................................................................................... 96 Figure 4.3.6 Sample of Physical Representation ....................................................................... 97 Figure 4.3.7 Sample of Symbolic Representation ..................................................................... 97 Figure 4.3.8 Sample of Coding Representation ......................................................................... 97 Figure 4.3.9 Sample of Input Error ............................................................................................ 101 xiii Acknowledgments I would like to express my deep appreciation to my research supervisor, Dr. Ann Anderson, and the esteemed members of my supervisory committee, Dr. Cynthia Nicol, and Dr. Deborah Butler. Their unwavering support, invaluable guidance, and constructive feedback have been instrumental throughout the research and writing process, playing a pivotal role in bringing this research to fruition. Dr. Anderson, I am truly grateful for the privilege of being your student. Your dedication to teaching and your profound commitment to your students have profoundly influenced me and enriched my journey as an educator over the years. If I were to compare you to an animal, it would undoubtedly be an owl, the epitome of wisdom. Your exceptional intellect and astute observational skills have consistently left me in awe during my time as your student. Your serene and peaceful demeanor mirrors the wisdom and insights of an owl, making those around you feel comfortable and at ease. This, I believe, is the captivating charm of a wise owl. I extend my thanks to the following scholars who have been an academic inspiration through their classes and the scholarly dialogues we have shared informally: Dr. Samson Nashon, Dr. Stephen Petrina, Dr. Marina Milner-Bolotin, and Dr. Yan Liu. I am also appreciative of Dr. Kwesi Yaro, Dr. Rowan Bernard, and all the teachers in the professional development meetings for their camaraderie and unwavering support. To Alan Jay and Ms. Kalie Wai Fong, I offer my heartfelt gratitude for your exceptional administrative and technical support throughout my tenure at UBC. xiv Lastly, I extend my warmest thanks to my parents in China for their boundless love and unwavering support. xv Dedication I dedicate this dissertation to my wife (April) and child (Kevin). 1 Chapter 1: Introduction This exploratory study seeks to shed light on young children’s experiences within robotics education. With decades of experience, as a parent, instructor and curriculum designer, within varied educational robotics settings, I have witnessed firsthand, albeit informally, young children’s self-guided engagement with mathematics as they solve robotics problems. As a mathematics educator and researcher, I am aware of the vast research literature on problem solving and the growing research into the role of technology in supporting children’s early learning of mathematics (e.g. Goodwin & Highfield, 2013) and mathematics and robotics specifically (e.g. Highfield, 2010). In this chapter, I identify an existing research gap, outline the research questions guiding my study and speak to my study’s significance. 1.1 Statement of Problem Mathematical problem solving has consistently played a significant role in the field of mathematics education. It is widely recognized as an essential component of both teaching and learning mathematics, leading to its integration into school curricula worldwide (Liljedahl et al., 2016). In Canada, while the Ontario Curriculum emphasizes problem-solving as the core of a robust mathematics curriculum (Ontario Ministry of Education, 2020), the BC Ministry of Education emphasizes the crucial role mathematics plays in problem-solving across different areas of life (B.C. Ministry of Education, 2016). Similarly, problem solving tends to be a key tenet of most educational robotics’ programs. Therefore, while a review of the expansive research into problem solving in mathematics education is beyond the scope of this thesis, 2 reviewing research literature related to problem solving within STEM, and more specifically within robotics contexts, was of particular relevance for the current study. To offer students alternative ways to develop and improve their problem-solving skills, many different technological tools and platforms have been introduced to classrooms, with the goal of fostering critical thinking, creativity, and collaboration among students (Highfields et al., 2008). More specifically, scholars such as Barak and Zadok (2009), Ortiz and Smith (2015), Ardito et al. (2014), and Papadakis et al. (2021) have argued that technologies such as educational robotics provide students with hands-on, experiential learning opportunities that allow them to actively engage in problem-solving within authentic, real-world contexts. Notably, studies have indicated the possibility of introducing computational thinking through robots and tangible interface programming in early childhood and primary education (Sullivan & Bers, 2015; Hudson & Baek, 2022). Furthermore, Baek et al. (2019) highlighted a significant improvement in 2nd graders' computational thinking abilities as a result of engaging in coding robotics activities. These efforts underscore the growing recognition of the importance of integrating robotics and computational thinking into educational practices to foster students' skills and competencies in these areas. Despite the potential benefits of robotics in education, its adoption has encountered challenges, as highlighted by Stokes et al. (2022) in their study on the integration of robotics in elementary schools in Ontario, Canada. Participants in the study identified various factors hindering wider adoption, including insufficient integration of robotics into the curriculum and assessment practices, limited resources, and inadequate professional development and support for teachers. 3 Teachers in the study often perceived robotics as a separate entity that competed for valuable curriculum time. Furthermore, the absence of a dedicated robotics curriculum in British Columbia until Grade 8 ADST (Applied Design, Skills, and Technologies) results in a lack of early exposure to robotics education, further impeding its integration into K–12 classrooms. However, non-formal educational settings like clubs, summer camps, competition teams, and after-school learning centers have taken the opportunity, and the responsibility, to nurture students' interest in robotics from a young age. These settings play a notable role in the robotics education landscape, offering alternative routes for learning and skill development. Given the positive connection between early mathematics achievement and later academic success (English and Kitsantas, 2013), and the potential of educational robotics in mathematical problem-solving (Highfield, 2010) and in computational thinking (Hudson & Baek, 2022), I argue that further research is needed to understand how students, especially those in the lower grades of elementary school, engage with problem-solving activities within after-school robotics programs. Gaining insights from non-formal out-of-school robotics programs may, in the long term, allow researchers and educators to optimize its role in mathematics learning and to expand robotics education from an extracurricular activity into mainstream K–3 classrooms. 1.2 Research Questions Coupled with the need for mathematics educators to better understand children’s engagement in robotics settings, I was personally motivated to carry out this research because of my many years as a robotics instructor observing first-hand the transformative potential of robotics education. By conducting this study, I sought to unravel the nuances in how students navigated mathematics 4 concepts and regulated their learning processes in the setting of an after-school robotics program. Understanding these processes is vital in enhancing pedagogical practices for afterschool programs in particular, but also with potential implications for in-school classroom settings. The ultimate objective is to provide meaningful insights that can positively impact educational practices, creating enriched learning experiences for students, particularly in mathematical problem-solving. The key questions guiding this inquiry are: 1. How do young children use mathematics as they engage in robotics activities? 2. What self-regulated learning opportunities could robotics afford as young children engage in problem-solving activities? 1.3 Significance of the Study In recent years, there has been a notable shift in mathematics education towards an increased emphasis on problem-solving skills and critical thinking (BC Ministry of Education, 2016). This shift is driven by the understanding that fostering problem-solving skills and critical thinking not only equips students for success in various fields, but also nurtures their ability to adapt, think creatively, and make informed decisions in an increasingly dynamic and interconnected global environment (NCTM, 2013). Along with this movement, evidence showcasing the efficacy of educational robotics in teaching mathematics has been collected in various educational contexts and at various levels (e.g., Gura, 2012; Williams et al., 2012; Christensen et al., 2015; Gomoll, 2016; Gyebi, 2017; Suárez-Gómez and Pérez-Holguín, 2020). Additionally, scholars have advocated for the cultivation of self-regulated learners who actively take charge of their learning 5 process, an approach that empowers students to develop essential skills such as critical thinking, problem-solving, and adaptability (Zimmerman, 2001; Nilson, 2013; Zimmerman & Kitsantas, 2014). Consequently, further research to investigate the potential impact of robotics’ experiences on the mathematics learning and self-regulation of primary students is needed. However, research on the impact of robotics on mathematical problem-solving and self-regulation, especially among young students in elementary school, remains limited. My study aims to fill this significant research gap by focusing on primary students’ participation in robotics problem-solving sessions within established after-school learning centers. By examining how young children incorporated mathematics into their robotics experiences and how they independently guided their own learning, this study will enhance our understanding of robotics in relation to mathematics education and offer valuable perspectives for future research endeavors and educational approaches. 6 Chapter 2: Literature Review The literature review presented here focuses on theory and empirical research related to mathematics education and the integration of robotics in math classrooms. It is divided into five segments, each aimed to provide readers with an understanding of the literature that informed and guided this study. These sections encompass: 1. Theoretical Frameworks: I explored the theoretical underpinnings of my research, which included an exploration of constructionist learning theory and self-regulated learning theory. 2. Mathematics Problem-Solving: The next section explores the background, strategies, and approaches associated with effective problem-solving in mathematics education, shedding light on its significance within the context of this research. 3. Technology Integration in Mathematics Education: In this section, I reviewed related prior research concerning the integration of technology in mathematics education. 4. Educational Robotics: In this final section, I explored the applications of educational robotics within various educational settings, with a specific focus on elementary schools. 7 Through these four sections, I aim to provide readers with an informed foundation for comprehending the theoretical and practical landscape that underpins the research questions and objectives of this study. 2.1 Theoretical Framework My research is grounded within constructionism, a theoretical framework that advocates tangible, hands-on experiences as catalysts for meaning generation and, consequently, learning. This perspective holds particular relevance in the context of educational robotics, where there is a notable emphasis on personalized experiences, active exploration, and the creation of tangible artifacts throughout the learning process. Additionally, I incorporate the concept of self-regulated learning (SRL) to study the active control young students exert over their own problem-solving experiences. These two theories complement each other, as both recognize that learning is not a passive endeavor, but rather, a series of active processes in which students actively shape their thinking. Constructivism, an influential educational theory, emphasizes the significance of hands-on exploration and the manipulation of physical objects to facilitate the development of new knowledge. In line with Jean Piaget's constructivist epistemological theory, constructionism extends constructivism by asserting that effective learning occurs when learners actively participate in crafting tangible, public artifacts (Papert, 1980). In the realm of constructionist theory, learners are not passive receptors of information; instead, they assume active roles in the learning journey. Learning is characterized by a gradual internalization of action that leads to the “construction of knowledge structures” (Papert, 1991, p. 1). While both constructivism and 8 constructionism share a common emphasis on the active construction of knowledge, constructionism emphasizes the use of technology as a catalyst that amplifies students' ability to engage in hands-on, experiential learning within authentic, real-world contexts (Papert & Harel, 1991). It is evident that many efforts to integrate technology into the classroom have been within the context of a constructionist framework. For example, the integration of technology in educational programs, as highlighted by Hmelo-Silver and Barrows (2006), Cotič and Zuljan (2009), Juliani (2015) aligned with the principles of constructionism. In the constructionist learning framework, students actively engage in dynamic processes, acquiring knowledge through hands-on exploration and tangible material manipulation. This approach, emphasizing creative experimentation, fosters the development of insights, with a focus on creating communal artifacts collaboratively. Papert (1980) asserted that in constructionist learning, students learn through creative experimentation and the collaborative creation of tangible products, drawing their own conclusions. According to Papert's constructionism, learning becomes authentic when learners design and produce tangible products, situating learning within these products. Papert argued for a physical and tangible component in learning, complementing cognitive aspects highlighted by constructivism. Ackermann (2001) asserted that Papert's approach to learning provides insights into how ideas are formed through cognitive learning. Hmelo-Silver et al. (2006) argued that constructionist approaches prepare students for problem-solving, critical thinking, reflection, and meaningful knowledge construction. The constructionist perspective, with its emphasis on tangible products, collaborative creation, and co-construction of knowledge, provides valuable insights into effective learning environments. 9 The constructionist perspective offers a shift in educational paradigms, emphasizing the situated and collaborative nature of learning. This perspective, as advocated by scholars Papert (1980), Gergen (2001), Alesandrini et al. (2002), and Resnick (2017), urged a departure from traditional subject-centered models toward a relational focus. Teachers, under this framework, are seen to transform into facilitators, guiding and coaching students in achieving their own goals. Chi and Wylie (2014) further advocate “constructive behaviors” in which learners do not simply intake knowledge but also generate additional outputs to the context. This concept underlines that true understanding and knowledge development emerge through meaningful interaction and the construction of tangible creations. This view of constructionist theory underscores the significance of relationships, collective experiences, and the active participation in constructing students’ knowledge. While constructivism and constructionism have gained popularity, some criticisms are associated with these theories. Scholars such as Kirschner et al. (2006) asserted that the minimally guided approaches often associated with constructivist methods disregard empirical evidence demonstrating the ineffectiveness of unguided instructions within learning contexts. In a parallel vein, Matthews (1993) argued that while constructivists emphasize knowledge rooted in personal direct experience, the genesis of scientific concepts frequently originates from abstraction that surpasses sensory impressions. These criticisms share a common ground centered on prioritizing measurable outcomes. While measurable outcomes are indeed significant in education, these critiques imply a potential risk of oversimplifying learning to a mere checklist of achievements. This approach may overlook the significance of learner engagement, meaningful interaction among learners, and the motivation driving their learning processes. For 10 the current study which focuses on children’s experiences rather than achievement per se, such criticism while duly noted is not overly concerning. In addition to constructionism, for this research I drew on theories that helped to inform understanding about what it looks like for students to actively take control over their problem solving process. Self-regulated learning theorist Zimmerman (2000) suggests that productivity in learning hinges not only on the availability of diverse cognitive strategies but also on the control of motivation and perseverance to tackle problems and apply these strategies effectively. The significance of self-regulated learning (SRL) in academic performance is widely recognized by scholars across various educational levels, from elementary school (Zimmerman & Kitsantas, 2014) to junior high school (Masubuchi et al., 2022) and university (Pérez-González et al., 2022). Schroeder and Lester (1989) argued that students' learning of mathematics is most effective when it is self-generated. This means that problem-solving requires learners not only to possess a sufficient set of problem-solving strategies but also to have the metacognitive control and self-regulation to exert the effort necessary to engage with the required cognitive activities. This combination enables them to achieve better outcomes in problem-solving activities. Schoenfeld (2008) argued that equipping learners with problem-solving strategies and metacognitive control or self-regulation can lead to improved outcomes in problem-solving activities. Studies (e.g., Tzohar-Rozen & Kramarski, 2014; Wang & Sperling, 2020; Tee et al., 2021) collectively emphasize the critical role of self-regulation processes in influencing students' academic achievement and mathematical reasoning at all ages. Interventions targeting metacognition, cognitive, and motivational aspects appear particularly effective in enhancing mathematics-related outcomes and fostering positive affective states. These studies underscore 11 the impact of self-regulated learning on a child’s emotional, motivational and cognitive engagement in problem-solving activities. Furthermore, models of SRL, as put forward by Zimmerman (2000), Pintrich (2004), and Butler and Schnellert (2020), enhance understanding of the web of influences on self-regulated learning. For example, these authors propose that learning unfolds through the dynamic interactions between individuals and learning contexts. In this view, learning is shaped (afforded; constrained) by the qualities of the situations students encounter. The integration of self-regulated learning principles in the context of robotics acknowledges the dynamic and interactive nature of learning, providing me a lens through which to analyze how students navigate and persist through the complex problem-solving challenges presented by robotics activities. Aligned with research on SRL, in the field of mathematics education, researchers have uncovered the importance of metacognition in children’s mathematical problem-solving (Schoenfeld, 1987; Silver, 1994). The metacognition framework, as proposed by Schoenfeld (2008), emphasizes the importance of learners’ metacognitive control, which extends beyond mere awareness and understanding of cognitive processes. The author argues that students must control, monitor, and self-regulate their thinking in order to solve problems effectively. In problem-solving, metacognitive control involves the deliberate and conscious regulation of cognitive processes, where learners not only recognize their own thinking but actively intervene to manage and optimize their cognitive activities. As such, metacognitive control aligns with the concept of strategic action within a Self-Regulated Learning (SRL) model. A child skilled in self-regulation, as underscored by Butler et al. (2017), not only comprehends how to navigate 12 through frustration in problem-solving but, more critically, has learned also how to anticipate and prevent undesirable feelings from arising in the first place. Given the compatibility of SRL and metacognition principles, I chose to focus on SRL, recognizing the complex interaction between cognition, metacognition, and motivation in the context of robotics activities. A closer examination of SRL in the robotics context offered an opportunity to explore intentional actions students undertake to monitor, control, and reflect on their learning experiences in these activities. Constructionism and SRL are connected through the idea that learners play an active role in constructing their knowledge and regulating their own learning processes. The constructionist perspective supports the notion of learners as active agents in their educational journey while constructing their knowledge. Concurrently, a SRL perspective suggests that learning is not a passive act of absorbing information, but rather a sequence of active processes focused on asserting control over one's own thinking (Perry et al., 2006; Vassallo, 2013). The synergy between constructionism and SRL provided me a more comprehensive understanding of how students participated in problem-solving activities with robotics. By integrating these two frameworks, I could examine not only the mathematical aspects in the problem-solving experiences but also explore how students regulate their learning processes within the unique learning environment. This included examining individual tasks, interactions with peers, and feedback mechanisms, thereby providing a more holistic perspective on students' engagement and learning in robotics activities. . 13 2.2 Mathematical Problem Solving Throughout the evolution of educational methodologies, the emphasis on cultivating problem-solving skills has endured as a cornerstone in mathematics education. Educational authorities highlight the pivotal role of problem-solving skills in fostering critical thinking, preparing students for real-world challenges, and enriching their mathematical capabilities. B.C. Ministry of Education (2016) and the National Council of Teachers of Mathematics (NCTM, 1996) aligned with this perspective, emphasizing problem-solving as a cornerstone in a robust mathematics curriculum. A subsequent update to the Principles and Standards for School Mathematics in 2000 reinforced the central role of problem-solving in mathematics education (Burris, 2005). Problem-solving is a multifaceted skill encompassing the ability to comprehend the environment, identify complex issues, review relevant information, devise and assess strategies, and implement solutions to achieve desired outcomes (Fissore et al., 2021). According to NCTM (2010), problem-solving in mathematics not only serves as a practical application of mathematical knowledge but also as a purposeful method for acquiring a deeper understanding of the subject. This aligns seamlessly with the constructionist approach and resonates with the principles of self-regulated learning (SRL). Both perspectives underscore the active engagement of learners in constructing their understanding of mathematical concepts, emphasizing that learners are not passive recipients but active participants in the learning process. Within the constructionist viewpoint, there's a recognition of learners as active agents in constructing knowledge and meaning through their interactions. Papert (1980) advocated for facilitating active learning by helping students “not only to learn but to learn about learning” (p. 177), a 14 sentiment in harmony with the SRL perspective that acknowledges the learner's active role in constructing mathematical understanding. As highlighted by Butler et al. (2017), problem-solving encompasses various dimensions, involving decision-making about when and how to apply specific strategies, setting goals, comprehending expectations, and adhering to rules. This multifaceted view underscores the dynamic and interactive nature of the problem-solving process. The holistic perspective on mathematics education, as presented by Cobb et al. (1991) underscores the profound impact of goal-directed problem-solving activities. From the perspective of Cobb et al. (1991), engaging children in goal-directed activities that involve problem-solving serves a purpose beyond simply finding solutions to specific problems. Through the problem-solving process, children develop operative mathematical concepts that are constructed through reflective abstraction (p. 187). This viewpoint, shared by scholars such as Schoenfeld (1988) and Boaler (2022), presents mathematics as an inquiry-driven subject that extends beyond facts and procedures. This viewpoint aligns with a broader understanding of mathematics education not merely as a collection of facts and procedures but as an inquiry-driven subject that cultivates an appreciation for discipline and nurtures adaptable minds. Schoenfeld (2016) claimed that problem-solving learning, as any other learning, is a culturally shaped and defined phenomenon. Individuals develop their understanding of any subject, including mathematics, through their engagement in a “community of practice” where that subject is actively applied and practiced. Polya (1971) placed significant emphasis on the role of teachers in the problem-solving process. He argues that merely presenting students with learning 15 material, even if it is well-constructed, does not effectively facilitate their learning. These studies highlight the importance of social interactions, cultural context, and participation in shaping one's learning experiences and comprehension of mathematical concepts. Along these lines, there is a growing advocacy for the integration of mathematical problem-solving as opportunities for students to apply mathematics in real-world scenarios. “School should focus its efforts on preparing people to be good adaptive learners so that they can perform effectively when situations are unpredictable and task demands change” (Resnick, 1987, p. 18). In contrast to methods that solely aim to transmit disconnected facts and skills, Tripathi (2008) underscores the importance of incorporating multiple representations in mathematical problem-solving. This approach involves employing diverse techniques that encompass numbers, symbols, written language, and visual aids to enrich students' comprehension and involvement. Duval (2006) further emphasized the significance of employing multiple representations in mathematics education, highlighting its essential role in enabling learners to leverage the distinctive benefits offered by each form of representation. These studies exemplify the holistic perspective championed by numerous scholars, viewing mathematics as a process of meaning-making that transcends mere procedures and skills. Boaler (2013) recommended that, instead of focusing on routine skills like multiplication or memorizing theorems, educators should present open tasks that enable students to envision greater achievements and opportunities for improvement. The perception of struggle and mistakes in the realm of problem-solving holds substantial influence over students' attitudes and engagement in mathematics (Boaler, 2022). The prevailing 16 notion that intellectual struggle signifies weakness can create a counterproductive classroom culture, deterring students from active participation (Stigler & Stevenson, 1992). To address this issue, Dweck (2008) suggested shifting the narrative around struggle and emphasizing the importance of perseverance and effort. As highlighted by Whitehead (2012), praise should be directed toward students who persist in the face of challenges, utilize multiple strategies, and exhibit commitment and hard work. This approach not only motivates students to confront errors but also nurtures a sense of ownership over their learning processes, promoting a growth mindset and fostering a resilient and empowered mathematical mindset (Butler et al., 2017). What these studies have in common suggests that acknowledging and embracing struggle and setbacks as an inherent aspect of the learning process can act as a catalyst for heightened engagement in mathematics and the cultivation of essential problem-solving skills. Guided by these studies, I explored the participation within robotics among young students, examining their perceptions of errors and challenges encountered throughout the process. In the subsequent literature review, I will continue the exploration of several models and processes rooted in constructionist and self-regulated learning (SRL) frameworks. By examining these frameworks, I aim to provide insights into how these models and processes can be effectively applied to contribute to my research. 2.2.1 Polya’s Problem Solving Model Polya views mathematics as an active and dynamic endeavor, likening it to the physical sciences in its reliance on guessing, insight, and discovery (1971). According to Polya, mathematics is not a passive process but an engaging activity that involves active exploration of problems. The 17 important role of engagement, particularly active participation in discovery, suggests that much of the learning in mathematics unfolds through a process of trial and error, educated guessing, and the development of insights. Polya's four-principle approach to problem-solving (Polya, 1971) suggests the following model (Figure 2.1) when solving a problem: However, teaching heuristics to cultivate the adaptive problem-solving skills Polya envisioned remains a challenging task (Schoenfeld, 1992). Problem solving, as envisioned by Polya, is about developing the ability to tackle new challenges without predefined solution methods. Many textbooks incorporate problem-solving sections that, instead of fostering genuine heuristic thinking, tend to reduce the process to mere algorithms through drill-and-practice methods (Schoenfeld, 2016; Boaler, 2022). I have observed that teachers in traditional settings often presented students with strategies, and asked students to practice them in repetitive exercises, and tested on their practiced algorithms. In this approach, the essence of Polya's heuristics, which involves grappling with novel and unfamiliar tasks, is lost. 2.1 Illustration of Polya’s 4-step Problem-Solving Model Note. Adapted from Polya, 1971. 18 Models that build upon Polya's work often follow similar phases, but with variations. Woods et al. (1975) proposed a problem-solving model consisting of five steps: defining the problem, thinking about it, planning a solution, carrying out the plan, and looking back. The “defining the problem” step aids in interpreting the information, identifying the system under study, and setting criteria for success. “Thinking about it” involves gathering knowledge and information. “Planning a solution” entails considering strategies and selecting the best one. “Carrying out the plan” involves execution, while “looking back” encourages reflection. Schoenfeld (1985) expanded Polya's approach by introducing a fifth phase in his problem-solving model. The additional phase pays attention to the processes of planning and exploring. Further, Schoenfeld emphasizes an essential distinction between routine and non-routine tasks in problem-solving. According to the author, routine problems align with well-established methods, lacking the need for creativity. On the other hand, non-routine problems demand a higher level of ingenuity and cognitive effort from students. This addition provides a valuable perspective for me to investigate the non-routine nature of robotics activities, recognizing the importance of strategic planning and the exploratory aspect within the process. In my study, a broader definition was incorporated, contrasting with an approach focused solely on procedural and pre-prescribed skills, I viewed mathematics as a domain where students’ discovery, exploration and representation could play roles in advancing understanding and knowledge. This broader perspective allowed for a more comprehensive understanding of mathematical problem-solving, aligning with the heuristic viewpoint of Polya (1971) and Schoenfeld (2016). 19 2.2.2 Self-Regulation Models Various Self-Regulated Learning (SRL) models have been developed by researchers, each aimed at understanding and explaining the components of self-regulated learning from diverse theoretical perspectives. Zimmerman, a key founder of research in Self-Regulated Learning (SRL), introduced a very influential model to depict the interactions influencing SRL (Zimmerman, 1986). Zimmerman proposed a cyclical feedback loop consisting of three phases: forethought, performance, and self-reflection. During the forethought phase, students engage in planning, goal setting, and strategy identification. During the performance phase, they execute their plan and use their strategies. During the reflection phase, they engage in self-evaluation, self-monitoring, and adaptation. Zimmerman (2000) underscores the significance of self-regulated learning (SRL) in younger students. His (2000) social-cognitive model has also consistently emphasized the role of self-efficacy, which refer to students’ situated appraisals of how likely they are to succeed in achieving goals in a particular context, as well as social and environmental factors on self-regulated learning. Winne and Hadwin (1998, 2008, 2013) recognized and expanded Zimmerman’s model by emphasizing students’ effort in the phase of monitoring and the use of (meta)cognitive strategies. Their SRL model foregrounds components like task definition and cognitive evaluation, which are specifically relevant to students' engagement in robotics activities as these activities feature complex problem-solving scenarios, requiring students to define tasks and engage in task appraisals to assess their understanding and approach. 20 SRL models have consistently recognized the role of social contexts and social interactions in self-regulated learning. In her work, Hadwin et al. (2018) explored more specifically what it looks like when students are learning together. She and her colleagues defined socially shared regulation as “a group’s deliberate, strategic, and transactive planning, task enactment, reflection, and adaptation” (p. 86). The author argued that by facilitating group-level planning, collaborative enactment of tasks, joint reflection, and adaptive adjustments, socially shared regulation has a potent mechanism for leveraging the diverse strengths and perspectives within a group. This notion of socially-shared regulation holds significant implications, particularly in understanding the collective intelligence of a group in robotics activities. In my study, the social component was integral as I investigated learning not only in individual settings but also in pairs. By studying problem-solving in pairs, I aimed to understand how students collaboratively engaged with robotics tasks, shared their understanding, planmed and executed strategies together, and reflected on their collective experiences. The examination of both individual and paired learning experiences allowed for a better understanding of how the social dynamics contributed to self-regulated learning in the context of educational robotics and problem-solving activities. Butler et al.’s (2017) model of SRL added a valuable sociocultural perspective by extending beyond individual cognitive processes, acknowledging the profound impact that the broader socio-culturally and historically situated environment has on students' learning experiences. The authors propose an integrative framework (Figure 2.2), underscoring the importance of contextualizing SRL within context (see also Butler & Cartier, 2018). The integrative model illuminates the intricate interplay between learners' engagement, emotions, motivation, the learning environment, and the various elements involved in the process of self-regulation. It 21 highlights the need to contextualize SRL within a comprehensive framework that considers diverse internal and external factors. The above SRL models foreground common components and processes involved in self-regulated learning. They generally include strategic, metacognitive, and motivational components that orchestrate learning processes within a particular domain (Zimmerman, 2000; Butler et al., 2017). This shared understanding underscores the dynamic and iterative nature of SRL, where learners engage in multi-dimensional aspects of the learning. This understanding is particularly relevant to my study, as the combination of internal and external elements within the context created a unique environment. In this setting, various elements such as the presentation of tasks, the facilitator's instructions, feedback in the activities, dynamics between partners, and Figure 2.2 An Integrative Model of Self-Regulation Note. Excerpt from Butler et al., 2017, p.8. 22 the assessment format collectively influenced students' engagement in these robotics activities. Taking a holistic view of mathematical problem-solving in my study allows me to comprehend these children’s use of mathematics, as well as the dynamics of SRL, within a robotics context, more broadly. 2.2.3 Computational Thinking Computational thinking is a problem-solving approach that draws on concepts from computer science to address complex issues in various domains. The Carnegie Mellon Center for Computational Thinking describes computational thinking as thinking algorithmically and with the ability to apply mathematical concepts such as induction to develop more efficient, fair, and secure solutions (Carnegie Mellon, 2023). It depicts computational thinking as a cognitive process that involves articulating concepts not only with sufficient clarity but also in a systematic manner. In analogy, it resembles a student to instruct a computer to execute his/her ideas. The process emphasizes that students approach problems in a systematic manner, and to develop solutions in terms that are simple and clear enough to be executed by a computer. Recognized as a foundational skill, computational thinking empowers individuals to approach and solve problems in a logical and systematic fashion. A growing movement supports the incorporation of Computational Thinking (CT) into Canadian curricula and, more broadly, into curricula worldwide (Wing, 2008; Larkin & Miller, 2020). The integration of CT and coding in K-12 education has also received considerable attention in British Columbia. The extent of such integration varies across school grades, ranging from elective courses in computer science to mandatory inclusion in the broader curriculum. In 2016, the BC Ministry of Education 23 announced curriculum changes aimed at teaching coding basics to students by the end of grade 9. In grades K-5, teachers are encouraged to integrate CT skills into other subjects with an emphasis on design thinking (British Columbia Ministry of Education, 2016b). These recent curriculum changes in British Columbia reflect a significant commitment to integrating CT throughout the province's education system. Wing (2008) conceptualizes computational thinking (CT) as a cognitive process that encompasses essential elements such as abstraction, decomposition, and heuristic reasoning. Her earlier work (Wing, 2006) emphasizes the strong connections between computational thinking, particularly coding, and mathematical thinking. Expanding on Wing's framework, scholars from the University of York (2023) provided a comprehensive characterization of CT by introducing four key techniques. These techniques include: 1. Decomposition: Involves breaking down a complex problem into smaller, manageable parts, facilitating a more systematic and approachable solution strategy. 2. Abstraction: Focuses on identifying the most crucial information and elements of a problem while filtering out irrelevant or unnecessary details, promoting a streamlined understanding of the core issues. 3. Pattern Recognition: Entails finding similarities and patterns within data or information, enabling individuals to recognize recurring structures that can aid in problem-solving. 24 4. Algorithm Design: Encompasses the process of creating step-by-step solutions, akin to a programmer developing instructions for a computer algorithm. This technique emphasizes the systematic formulation of solutions to complex problems. Studies have consistently demonstrated the positive impact of utilizing technologies on students' computational thinking skills. For example, Sung et al. (2016) found that children using tablets exhibited processes similar to mathematical and engineering thinking. Sáez-López et al. (2016) found that using block-based programming language in social sciences and arts courses improved students' computational thinking. Noh and Lee (2020) found that teaching programming through robot activities enhanced fifth and sixth grade students' computational thinking and creativity. Atmatzidou and Demetriadis (2016) found that training middle and high school students to use the Lego Mindstorm educational robot kit improved their computational thinking. The students showed significant improvement in various sub-factors of computational thinking, including generalization, algorithms, modularity, and problem decomposition. Chen et al. (2017) conducted a study focusing on 121 grade five elementary students who were using educational robots. They found that the robotics-based curriculum improved students' computational thinking skills, particularly in text-based and visual programming. These studies demonstrate that engaging with these technology tools provides learners with valuable experiences that contribute to the development and enhancement of their computational thinking abilities. The hands-on and interactive nature of programming and robotics activities fosters a range of cognitive processes associated with computational thinking, such as problem decomposition, pattern recognition, abstraction, and algorithmic design. 25 2.2.4 Engineering Design Process The Engineering Design Process (EDP) represents an alternative approach to problem-solving, particularly in the field of engineering. It is a systematic and iterative methodology used by engineers to develop solutions to problems. As suggested by Van Meeteren et al. (2010), the engineering design process isn't something foreign or external to young students’ learning experience; rather, it's an inherent and integral component, particularly in high-quality early childhood education programs. The key here is the synergy between the engineering design process and educational philosophies rooted in constructivism. Chin et al. (2019) emphasized that both learning about design and through design have the potential to enhance learning outcomes and problem-solving skills that extend well beyond the school years. Figure 2.3 Engineering Design Process Diagram Note. Excerpt from Mangold, 2013, p.6. 26 This educational approach emphasizes active learning, where students construct their understanding of the world through hands-on experiences. Atman et al. (2007) delineated ten stages within the engineering design process: (1) problem definition, (2) information gathering, (3) idea generation, (4) modeling, (5) feasibility analysis, (6) evaluation, (7) decision-making, (8) communication, (9) implementation, and (10) design revision. In a similar move, Mangold (2013) introduced a modified 8-step engineering design process, as shown in Figure 2.3. Its overarching goal extended beyond the mere transmission of knowledge; it aimed to foster a holistic educational experience that seamlessly blended theoretical understanding with hands-on, real-world applications. By doing so, the module was intended to bridge the divide that often exists between abstract classroom learning and practical, everyday life. As per Mangold (2013), EDP involves a systematic approach to problem-solving, incorporating analysis, creativity, and refinement in the pursuit of effective and efficient solutions. The 8-step module was implemented in two 7th grade math and two 8th grade science classrooms over the course of two years. One of the primary goals of the program was to actively involve middle and high school students in mathematics and science through engineering projects. These projects aimed to enhance students’ understanding of fundamental concepts in both math and science through hands-on activities. The author claimed that the implementation of EDP not only helped improve students' problem-solving abilities but also served as an effective means to introduce them to various engineering disciplines. The above literature review looked into the spectrum of problem-solving techniques and models. Whether it's mathematics, engineering, self-regulated learning, or any other field, the 27 overarching concept involves moving from interpreting a problem to generating potential solutions and finally assessing the outcomes. Together, these models provide a framework for understanding a student’s systemic problem-solving approach in robotics. 2.3 Technologies Integration in Mathematics Education The exploration of technology integration in mathematics education reveals a complex landscape shaped by historical perspectives and evolving trends. Cuban (1986) summarized the history of technology integration in American classrooms to 1985, from the use of instructional film clips to the rise of computers, and concluded that these technologies failed “the promise implied in these (technological) aids [that] caught educators’ attention: individualized instruction, relief of the tedium of repetitive activities, and presentation of content beyond what was available to a classroom teacher” (Cuban 1986, p. 4). Cuban urged those who had simply encouraged students to use computers and the internet to invest more time into training teachers to take advantage of technology in classrooms. Young et al. (2018) conducted a systematic review of meta-analytic research to examine trends in prior research on effectiveness of using technology in mathematics classrooms, technology which included computer-assisted instruction, calculators, mathematics software, and mobile technology. They found that the affordances and functions of technology and its alignment with instructional goals are key to effectively integrating technology in mathematics classrooms. Rakes et al. (2020) conducted another systematic review of meta-analytic research to examine 28 trends in prior research on effectiveness of using technology in mathematics classrooms. Rakes et al. organized educational technologies into five categories: (1) calculators, (2) probe ware (e.g., data collection devices), (3) software, (4) hardware (e.g., computer), and (5) the Internet. They determined that the available evidence is insufficient to make conclusive statements regarding the overall effectiveness of technology in enhancing mathematics learning outcomes. While earlier technologies faced skepticism, the advent of newer technologies such as mobile devices, applications, interactive learning platforms, and educational robotics has rekindled interest in education. The focus is on technology that promotes multisensory engagement, hands-on participation, collaborative learning, problem-solving, and construction activities. According to Ntuli (2015), these newer technologies have further advanced the adaptability and interactivity of technology-enhanced learning. Ntuli argues that the technologies that enable personalized learning experiences have the capability to customize content based on students' current knowledge and accommodate their individual interests and preferences. These technologies, through the use of immersive tasks and the promotion of collaborative environments, offer students the chance to actively engage with their surroundings, share ideas, and build upon their existing knowledge and skills through interactive experiences. Others argue that these newer technologies also have the potential to increase student engagement. (English and Halford ,2009; Docan-Morgan, 2015). Their studies claim that they have observed various forms of engagement, including behavioral and emotional engagement as students use touch screens on tablets, construct robotic components, or actively participate in 29 interactive whiteboard activities. Gura (2012), Christensen et al. (2015), James (2016), and Gomoll (2016) argue that technology can be used as a means to ignite students' interests, increase their participation, and enhance their engagement in the learning process. Peters (2007) highlights the evolving landscape of education in the digital age, allowing students to actively seek personalized and engaging experiences tailored to their unique information requirements. As technology continues to advance, its integration into mathematics education holds promise for enhancing not only the delivery of content but also the overall learning experience. 2.4 Educational Robotics in Mathematics Education The potential of educational robotics has extended into the realm of mathematics classrooms. The initial endeavor of such integration can be traced back to the 1960s and 1970s. One notable example is the LOGO programming language and its associated Turtle graphics, developed by Seymour Papert and his colleagues at the Massachusetts Institute of Technology (MIT) in the late 1960s. Over the years, block-building systems that originated from Papert's original LOGO project have undergone transformations and evolved into programmable construction systems, giving rise to the educational robotics programs we witness in schools today. Educational robots now come in various types distinguished by their appearances, structures, and functions. These features have significant implications for curriculum design, instructional activities, and learning objectives (Benitti, 2012). The BC Ministry of Education’s definition of educational robots as machines capable of autonomously executing complex actions (B.C. Ministry of Education, 2016) assumes a learning environment that involves assembling and coding these robots. 30 Earlier studies on robotics in education (Papert, 1980; Bransford, 2000; Highfield, 2010) indicated that working with programmable robots engage students in exploring spatial concepts, using problem-solving skills, and practicing metacognitive processes. Blanchard et al. (2010) conducted a case study, focusing on robotics-based learning in a local elementary school. The authors noted that robotics-based learning could create creative and inclusive environments enabling students to immerse themselves in interdisciplinary exploration. The authors agreed with Papert (1999) that robotics learning affords opportunities for students to engage in problem-solving activities to actively explore, observe, and discover discrepancies between prior knowledge and current experiences and therefore to construct new knowledge by working with digital and physical artifacts. Evidence demonstrating the effectiveness of educational robotics as a tool for teaching mathematics topics has been gathered in various educational settings and at different levels, including higher education (e.g., Gyebi, 2017; Suárez-Gómez & Pérez-Holguín, 2020), high school (e.g., Williams et al., 2012; Gomoll, 2016), and elementary school (e.g., Gura, 2012; Goodwin & Highfield, 2013; Gagliardi et al., 2021; Hudson & Baek, 2022), as well as both formal classrooms and non-formal settings such as after-school STEM programs (e.g., Christensen et al., 2015). Greenberg et al. (2019) provides insights into various ways middle to high school students can utilize algebra, data analysis, and trigonometry with typical educational robots. Ortiz (2015) revealed that a short yet intensive extracurricular learning program, using problem-solving in engineering design with robotics materials as the real-world context, led to significant student progress in specific mathematics topics. Atmatzidou and Demetriadis (2014) found that the modularity of robotics afforded middle school students with the opportunity to 31 construct an understanding of complex challenges and to develop problem-solving skills. Their research showcased some of the students' responses regarding their progress in problem-solving: "Now I approach problems differently and find solutions more readily" and "I've altered my problem-solving approach" (p. 48). These responses highlight the tangible impact of the robotics activities on students' problem-solving skills, indicating growth and development in their ability to approach and resolve challenges effectively. Even though there is a scarcity of studies specifically targeting young children's mathematical problem-solving skills, there are still valuable insights to be learned from existing research on robotics. For example, scholars (Sullivan & Bers, 2015; Hudson & Baek, 2022) revealed that even at a young age, from pre-kindergarten to Grade 3, children are capable of acquiring and mastering fundamental robotics and programming skills by devising creative solutions to complex problems presented to them. Foss et al. (2019) also found that instructing students about robotics had a positive impact on both their academic performance and behavioral development. A body of research underscores the use of robotics as a means to captivate children's interest in problem-solving (Papert, 1980; Sullivan & Bers, 2012; Goodwin & Highfield, 2013; Gagliardi et al., 2021). While there is immense potential in integrating robotics into math education, there are significant challenges in ensuring a seamless connection between these disciplines. Savard and Highfield (2015) conducted a study titled “Teachers’ talk about Robotics: Where is the Mathematics?”, concerning the integration of mathematics with robotics activities in elementary classrooms. The study focused on teachers’ discussions after these activities. The study revealed insights into the 32 limited depth of pedagogical knowledge among teachers. It further revealed that the opportunities to enhance the usage of mathematics in the context of robotics were often unclear and constrained. The findings highlight the crucial role educators play in unlocking the full potential of robotics for enhancing mathematical understanding. Similarly, English (2016) argued that the integration of mathematics into multidisciplinary activities is an often-overlooked aspect. By reporting STEM integration in curriculum documents and policy reports, the author noted a deficiency in substantial research that offers evidence of the intended learning outcomes. The study indicates that mathematics learning seems to derive fewer benefits compared to other disciplines such as science in programs that purportedly prioritize STEM integration. Chuang and Cheng (2022) further emphasized that, despite the popularity of robotics programs, the primary focus often remained on teaching programming skills and robotic control, with insufficient alignment with math standards. Addressing these challenges in integrating robotics in education undoubtedly requires more studies in exploring and leveraging the possibilities that robotics presents for children relating mathematics. My study focuses on investigation of ways that young students used mathematics and the self-regulation within after-school robotics programs, aiming to identify potential opportunities that can be deliberately integrated across various educational contexts. By recognizing these prospective opportunities, the study can help educators better craft meaningful and enriching experiences for students in mathematics education. 33 Chapter 3: Methodology Researchers adopt case study designs when aiming to develop a comprehensive understanding of a specific entity within its authentic context (Yin, 2014). These entities, referred to as “cases,” are typically described as bounded systems representing some form of unity, such as individuals, places, phenomena, or social units (Merriam, 1998). Case study designs are instrumental in investigating a bounded system holistically, allowing researchers to collect, assemble, and relate various types of evidence. Thus, case study methodology seemed fitting for my qualitative study of children’s mathematical and self-regulating experiences within a particular robotics program. The utilization of case studies is a common and valuable research design in the field of educational robotics. Several researchers have employed this approach to explore self-regulated learning (SRL) in robotics, as well as children's mathematical experiences within the field. Given these precedents, it appears to be a suitable methodology for my study. For example, Bharatharaj et al. (2023) implemented a case study to gain insights into preschoolers' interpretations and reactions to delay-of-gratification aided by robots. McWhorter (2008) used a case study to explore students' motivations involving robotics. In a similar vein, Demetroulis et al. (2023) investigated two cases to study SRL strategies and mastery of computer courses using Lego robotics in a collaborative setting. Additionally, the case study design has proven useful for providing insight into young students’ mathematics and problem-solving skills in the context of educational robotics. For instance, Jurado et al. (2020) conducted a case study of STEM skills, including logical-mathematical and 34 problem-solving abilities, in four classrooms with 75 students in Spain. Cherniak et al. (2019) investigated a case and found that the process of children’s problem identification in robotics was dialogic, not only with peers and teachers but also with materials. Brender et al. (2021) investigated the role of educational robotics in formal mathematics education, with a focus on geometry for 15-year-old students. The focus of my study centered on a case of six students enrolled in robotics programs across three distinct learning centers within the Lower Mainland of British Columbia. These six students, in grades 2 or 3, were organized into four subcases. Two subcases consisted of a pair of students, and the remaining two subcases involved individual students. Treating each pair of students as a subcase offered an opportunity to explore problem-solving experiences within a collaborative dynamic. The selected learning centers shared a common curriculum, providing a consistent educational framework for all participants. The established organizational facility of the robotics programs naturally established boundaries for the study. The subcases were enclosed within the confines of the robotics programs and embedded in the overall student population in its nature setting. While the subcases in this study didn't meet the immediate phenomenon criteria, the incorporation of diverse and uncontrolled variables in the case study design facilitated a more comprehensive exploration of the intricacies of self-regulated learning (SRL) and problem-solving within authentic after-school settings. As highlighted by Creswell (1998), Yin (2014), and Butler and Cartier (2018), case study designs are particularly advantageous for observing complete cycles of learning and development unfolding in real time, taking into account 35 contextual factors and enacted pedagogical practices in naturalistic settings. In formulating my case study approach, I found valuable guidance in the work of Yin (2014). His insights and methodologies provided a solid starting point for my research. One prerequisite for conducting successful case studies is that they take place within a well-defined system (Yin, 1994). For the current study, I observed six students’ participation in individual or paired work on pre-designed robotics activities, documenting a total of 24 activities unfolding within the naturalistic context of an after-school robotics program, bounded by time and geography. In addition, this case study of children’s use of mathematics and self-regulation within robotics was framed by theories of constructionism and self-regulated learning. 3.1 Settings The conducted research took place in three well-established learning centers, located in three distinct cities. Each center was situated in an urban area, within a 40–50-minute driving distance from the others. These locations allowed for geographic diversity, while supporting timely accessibility for my visits throughout the week. These learning centers operated throughout the year, providing after-school programs during weekdays and weekends. Enrolling in these programs required parents to make advance tuition payments, reflecting their financial situation and commitment to the sustained involvement of their children in the programs. These learning centers provided a range of STEM (science, technology, engineering, and mathematics) programs tailored to students in grades K-8, such as robotics, 3D-modeling, and video game designing. The demographic makeup of enrolled students predominantly consisted of 36 learners spanning grades K to 3 at each site. Robotics programs had consistently been the most popular choice among students. Among various robotics kits tailored for different age groups, LEGO WeDo robotics kits (version 1.0) emerged as the most popular choice within these centers for students in grades one to three, aligning with the specific focus of my study. The curriculum choice, particularly the use of LEGO WeDo robotics kits, was driven by the program's decision to align with the students' age group and their interest in robotics. While the connection to the provincial curriculum was not explicitly stated, the hands-on and problem-solving nature of the activities inherently involves mathematical concepts (i.e. geometry, measurement, patterns) contributing to the students' learning experiences. Notably, the classrooms within these centers adopted mixed-grade and mixed-curriculum configurations. Each classroom at the learning centers featured four workstations (Figure 3.1) with computers and building elements that were shared among students. The robotics curriculum was accessible by clicking the icons on the computer screen. Each computer station enabled collaborative work for up to two students per station (Figure 3.1). This arrangement allowed students in different programs to work simultaneously. Figure 3.1 A Classroom and Workstations 37 The assortment of building elements and tools, including diverse geometric shapes and electronic components, offered students a broad spectrum of options. For example, angle connectors and gears (Figure 3.2) could be used to craft structures with unique angles, and electronic components such as a motor and a USB hub further allowed students to explore programming concepts, such as adding movement or enhancing overall functionality. In the robotics sessions, students participated in a series of pre-built activities. The flexibility of the curriculum allowed students to choose activities in sequence. In each classroom, there was a dedicated instructor. Each instructor was responsible for four students within their enrolled programs (i.e. robotics, 3D modeling). Since instructors aimed to encourage independent problem-solving, minimal assistance was provided. That is, instructors helped students by clarifying concepts, allocating learning materials (i.e. building blocks), providing assistance with the technical aspects of coding, and fostering collaboration among pairs of students. It is noteworthy that there was no formal assessment built into the instruction and no marks were assigned. However, students received quick, situated feedback from the robots themselves (and from each other, when in pairs) to support their learning. The popularity of the robotics program, coupled with the unique classroom dynamics, provided a rich research landscape. Figure 3.2 Samples of Special Building Elements 38 3.2 Participants The case of this study comprised six students participating in the robotics program across these three centers. I began my study by extending invitations to children who were enrolled in robotics programs in the three learning centers. Given the locations of the learning centers across various cities, each with its own unique operational schedules and availability of students, I used a purposeful selection process to recruit participants for this study. In particular, I asked managers at each center to send letters to parents of children who had been enrolled for three months, and would continue to be enrolled, for at least three more months. Six parents consented to have their child take part in the research. Coincidentally, two children in each of two centers attended the program at the same time, while a third child in one of those centers arrived at a different time and the other child attended a different center. To facilitate the management of the study and create an organized framework for data collection, participants who arrived at the same time each day were asked to work in pairs as a single subcase, thereby reorganizing the six participants into four subcases. The children appeared to be quite content when asked to work with a partner. On certain occasions, pairing occurred naturally in these learning centers when there was an unexpectedly higher turnout of students for the day. Thus, working with a partner was not an unfamiliar practice. In terms of this study, this organizational strategy allowed for a more efficient allocation of resources such as interview and observation time, as well as my weekly transportation to the three locations. Additionally, pairing participants created a dynamic context where collaboration and interaction could naturally unfold. All four children in the two pairs who were given the option to work together on robotics building activities and problem-solving tasks readily accepted and were comfortable 39 remaining together for the duration of the study. Conversely, participants who attended the study on their own schedules worked alone. Students in each subcase had prior exposure to robotics activities, with a minimum of three months of experience. This shared foundation in robotics ensured a baseline familiarity with the tools and concepts, allowing the study to focus on their engagement of problem-solving. Before commencing the formal study, I visited each center during drop-off time, taking the opportunity to introduce myself to instructors, parents, and the child participants. This informal orientation aimed to create a positive rapport. During these informal visits, valuable insights were gathered. All parents and students expressed a general enthusiasm in robotics. Four out of the six parents shared positive feedback about their children's performance at school and at home. The fifth parent indicated her child was spending a lot of time with his soccer team and other non-academic activities. The sixth parent mentioned that their child was excelling academically, particularly in mathematics as the child was learning two grades higher from a private tutor. This initial interaction, albeit brief, suggested all six students were interested in robotics and the parents perceived a positive influence of robotics on their children's overall performance. The students in each learning center exhibited a diverse range of cultural, ethnic, and educational backgrounds. The composition of the students in my study reflected the demographics of the respective learning center. Since the focus of the study was the children’s participation in the robotics program, I did not collect specific background details of the participants. Nonetheless, the diverse composition of students, spanning various cultural and ethnic backgrounds, contributed to the representativeness of the research. 40 Throughout the study, the selected children continued to engage in their usual learning activities, following the curriculum of the program in which they were enrolled. My formal interactions with the students took place during their regular after-school hours at their respective learning centers. During the week immediately following the orientation, I commenced the study by visiting each of the three centers on designated days over a period of seven weeks, coinciding with each child's scheduled session at their respective center. These sessions generally ran for 50-60 minutes. The interviews extended beyond the regular hour, with each visit lasting approximately 1.5 hours. Table 3.1 provides details on the participants, including their pseudonyms, ethnicities, grade levels, ages, as well as information about the location, meeting day, and time of their individual robotics programs. Individual (S) or Pairs (P) Name Grade Age Center Day Time Subcase 1(S) Sylvia 3 7 A Wednesday 4-5pm Subcase 2(S) Tania 2 8 C Friday 4-5pm Subcase 3(P) Carl & Princeton 2 7 B Thursday 5-6pm Subcase 4(P) Jason & Kent 2 7 A Saturday 1-2pm Table 3.1 Summary of participant information 41 3.3 Robotics Activities The focus of this study remained on the case of six children engaging in robotics sessions as they normally would. The robotics activities were organized by each center within a shared curriculum adopted by these centers. The curriculum, supplied by a local STEM educational company, accessible on all computers across all locations, facilitated a uniform learning experience. The curriculum consisted of three pages, each showing 12 icons, as illustrated in the screenshot (Figure 3.3a). Every icon on the page was associated with one robotics activity that was comprised of two primary stages known as the “Initial Assembly” and the “Challenge”, distinguished by a pair of clickable buttons displayed on the screen. For example, in the Butterfly project, clicking the first button (Figure 3.3b, the arrow icon marked by A) directed students to the instruction of Initial Assembly. During this stage, step-by-step instructions were provided for the required building parts and their placements. The screenshot displayed the 8th step, which marked the completion of constructing the swivel Figure 3.3a An Example Page of Robotics Curriculum 42 structure (Figure 3.3b, left). Guided by the step-by-step instructions displayed on their screens, students assembled robot models, laying the foundation for later modification and development. Having completed the initial assembly, students progressed to the next stage by clicking the second button (Figure 3.3b, the arrow icon marked by B on the right). It transitioned students into the domain of problem solving, a core element of the robotics sessions. This transition led them to a new stage, termed “Challenge” by instructors, in which additional tasks were presented as shown in Figure 3.3b, right. As shown in the screenshot, the instruction of the challenge was displayed on the screen, challenging students with a task to enhance the initial assembly to replicate a movement of a fluttering butterfly. This type of challenge involved refining the initial assembly and programming it accordingly. During the Challenge stage, no step-by-step instructions were provided. Figure 3.3b Screenshots of Two Buttons Associated with Butterfly Icon 43 It is worth noting that some activities presented options for challenges (e.g., in the Clock/Windmill activity, two options were provided). Additionally, while the phrasing of the challenges associated with the second button varied across three centers, it consistently followed the same format. Instructions for this stage were typically conveyed through text descriptions, with no specific building instructions or coding samples provided. While completion of both stages of the robotics activities typically took place in one room with one or two students working on one computer station, as shown previously in Figure 3.1, my study required a minor change for research purposes. Upon completion of the initial assembly, the participating children moved with me to another room nearby. Similar to other rooms, in this room students had access to robotics kits, educational materials, and relevant tools necessary for engaging in hands-on activities. When transitioning to the other room, students carried with them the completed initial assembly to continue to advance and broaden their projects further. In this “Challenge” stage, the instructor was not physically present in the room with the study’s participants. Rather, I assumed the dual role of researcher and instructor, whereby I addressed the needs and queries of the students if they arose. The movement to another room was intentional, offering enhanced privacy conducive to interviews and enabled me to capture screenshots of their coding scripts for subsequent analysis. Even though the environment changed, students continued to stay focused on the task at hand, as depicted in Figure 3.4, wherein a pair of students were engaging collaboratively in the problem-solving stage (“Challenge”). My in-situ observations, as well as my analysis of data, suggest that relocating to a different room had no noticeable impact on the students’ behavior. 44 Given the individual progression rates through the curriculum, coupled with the anticipation of two students graduating earlier than the others, I documented whichever six activities each child chose over the two-month period while all participants were still enrolled. Throughout the study, the six students collectively participated in a total of 11 distinct activities, with each student individually engaging in six of these activities. Table 3.2 offers an overview of the activities in which each student took part. The descriptions of each activity provided in Table 3.2 consolidate information from the readings on the students' screens (See Appendix A for details) and the briefs provided by the instructors to the students at the start of the lessons. Additionally, the table features photographs of students' sample works, either captured using a cell phone in the classroom setting or obtained from video recordings during the Challenge stage. Instances where an activity presented students with a choice between two different options (e.g., clock/windmill) are indicated accordingly. Figure 3.4 Area for Challenge Stage and Interviewing 45 Since Tania's clock activity marked the initial session observed, the listing of activities follows the order of Tania's participation. Consequently, as depicted in Table 3.2, the sequence in which Tania (and the other children) selected their activities did not align with the progression of the icons displayed in Figure 3.3a, from left to right or top to bottom. Sample Works Description of Challenge Participants Session and Site Windmill/Clock Build a stand and choose between modifying it into a clock or a windmill. Tania (Clock) Sylvia (Clock) Carl and Princeton (Windmill) Jason and Kent (Windmill) Session 1, Center A Session 1, Center C Session 5, Center B Session 5, Center A Tower/Crane Build and code a tower that reaches a height or build and code a crane that swivels. Tania (Crane) Jason and Kent (Tower) Tania (Tower) Session 2, Center A Session 4, Center A Session 5, Center A 46 Sample Works Description of Challenge Participants Session and Site See-saw/Hammer Build and code a seesaw that goes up and down or a hammer that hits a nail. Tania (See-saw) Carl and Princeton (See-saw) Sylvia (Hammer) Session 3, Center A Session 2, Center B Session 6, Center C Walker Build and code a robot to climb up a ramp or go a certain distance. Tania Carl and Princeton Jason and Kent Session 4, Center A Session 4, Center B Session 1, Center A Frog Build a frog that makes an interesting sound when it eats. Tania Session 6, Center A Butterfly Build and code a butterfly that flaps both wings in a pattern. Sylvia Session 5, Center C 47 Sample Works Description of Challenge Participants Session and Site Bulldozer Build and code a bulldozer to pick up LEGO blocks from a pile. Jason and Kent Carl and Princeton Sylvia Session 6, Center A Session 6, Center B Session 2, Center C Dragonfly Build a dragonfly that hovers on a geared pole. Jason and Kent Carl and Princeton Session 2, Center A Session 3, Center B Catapult Build and code a catapult to throw dice and produce the largest possible number. Sylvia Session 4, Center C Carousel Build and code a four-seat carousel to spin at an interval period of time. Jason and Kent Session 3, Center A 48 Table 3.2 Activities and Sites 3.4 Researcher’s Role As a parent, my journey into the world of robotics education began with my son's involvement in a school board enrichment program. He used a particular robotics kit to solve a maze problem. After the conclusion of the program, I continued to teach him at home. Witnessing his enthusiasm and engagement in these activities at school and at home sparked my interest in understanding the dynamics that underlie such experiences. After this initial exposure to educational robotics, I progressed from teaching my six-year-old to guiding a group at our local library, and eventually had an opportunity to instruct 20 third-grade children at a local elementary school. Years of closely observing these young learners led me to identify a research gap: a lack of exploration into how educational robotics influences mathematical problem-solving, particularly among elementary students. Sample Works Description of Challenge Participants Session and Site Sensor car/Bug car Build a car or a bug with wheels that can move to a spot. Carl and Princeton (Option: Sensor car) Sylvia (Option: Bug) Session 1, Center B Session 3, Center C 49 My ambition to explore the specific aspects of students’ participation in robotics was born from a combination of personal and professional experiences. I hold a belief that understanding these processes could offer insights for educators, curriculum designers, and policymakers, to optimize instructional strategies and create more enriching learning environments for young students. To bridge this gap, I embarked on a research journey to explore the potential of robotics activities to foster learning in mathematics. My prior work experience with an educational company provided me with a significant advantage when conducting this study. The company specializes in offering management services and curriculum support to learning centers, and camps. During my years teaching at libraries, learning centers, and schools, I gained an in-depth understanding of their operational dynamics. The curriculum I had used in these settings shared similarities with the curriculum utilized in the learning centers I was studying, providing me with insights into various aspects, including instructional content and the structure of the robotics program. In this study, my role was that of an observer and interviewer, with a focus on diligently documenting and analyzing the students' ongoing activities in the robotics context. None of the instructors, children, or parents in the centers involved in this study were familiar with me before the study commenced. During the initial assembly stage of the robotics activity, I positioned myself nearby to avoid interfering with students’ engagement and interactions. During the “Challenge” stage, I took on a dual role of researcher and instructor. Without the instructor present in the room, when students' queries arose, I emulated the guidance typically provided by instructors. For instance, when confronted with a coding obstacle, instead of providing a ready-50 made solution, I would offer hints, ask probing questions, or suggest alternative avenues for exploration. This approach not only aligned with the principles of constructivist learning, but also facilitated a more comprehensive exploration of the students' thought processes and actions. To maintain reflexivity and minimize any pre-existing beliefs that could influence the study, I regularly consulted with my advisors. This ongoing dialogue played a crucial role in safeguarding the study's integrity by ensuring that my perspective remained open to a range of viewpoints. It helped me stay vigilant and reduce any unintentional impacts on how the participants experienced the study. 3.5 Data Collection Over the course of two months, I observed a total of 6 sessions for each student or each pair of students' robotics activities, resulting in a total of 24 sessions. I used various methods to collect data. The combination of observation and interviews produced written fieldnotes, video and audio recordings, screen shots, and artifacts, thereby providing a rich corpus of data for understanding students' use of mathematics and SRL during robotics activities. 3.5.1 Observation Observational field notes (See an example in Appendix C) were made during both the initial assembly portion of activities (about 20-30 minutes) and the problem-solving sessions (about 20-30 minutes). Even though video-taping the building portion might have provided beneficial data on students’ mathematical thinking and SRL, it was logistically difficult to record only participants, since there were other students in the same space studying different programs at the 51 same time. Since building materials were shared among students, other students would have entered the recording scene when they moved around in the classroom. Audio recording was possible, but it was difficult to selectively identify the conversations of my participants on a recording containing multiple voices. I thus documented the building portion of the robotics sessions only through written field notes. The purpose of my observations was twofold: (1) to understand the ways students used math in robotics activity, and (2) to explore the SRL opportunities provided to students within these robotics practices. I therefore focused my field notes on key moments when students were mathematically engaged in problem-solving processes; I monitored students' interactions and recorded indications of strategic behavior of self-regulation. 3.5.2 Interviews: Problem-solving, Video Recall During each activity, students initially assembled a robot in the regular classroom. Subsequently, they relocated to a private room to partake in a problem-solving activity involving the same robot. Students were provided with a laptop for programming their robot. Two methods of video recording were employed during this session: one through the laptop's webcam and another via a smartphone. The laptop utilized dual-capture software to concurrently record both the students' coding processes and their close-up actions in front of the webcam of the laptop. Additionally, a digital camera was pre-set on the desk at an appropriate height to capture the facial expressions and body movements of the participating student(s). 52 Participants were informed about the recording process and gave their consent. To minimize the camera's impact, I positioned it discreetly, avoiding direct interaction with the participants. These efforts to maintain a respectful and non-intrusive setup appeared to minimize any undue influence on their actions and interactions. Videotaping was conducted using the camera while I simultaneously recorded field notes to capture additional details and observations. Typically, this problem-solving session lasted for 20-30 minutes. Following the completion of the problem-solving tasks, the students took a brief break from their work. During this time, I prepared one or two video clips extracted from the dual-capture recordings on the laptop for use in the forthcoming video recall interview. These video clips contained their coding scripts, close-up views of their faces or robots, and their voices. The intention behind using these clips was to assist the students in initiating discussions, recalling their thoughts, and reconstructing both the problem-solving process and their self-regulated actions. Subsequently, I conducted a 10–15-minute video recall interview session, using semi-structured questions and in-situ questions (Appendix D). The semi-structured questions, while minor in variation, served as a systematic tool to help students articulate their thoughts, strategies, and reflections on their robotics activities. In-situ questions were incorporated at times, allowing for spontaneous inquiries based on the unfolding dynamics of the robotics activities, adding depth and immediacy to the interview process. The interview was both videotaped and audio recorded. By integrating video recordings, screen-captured coding scripts, field notes, and interviews, a comprehensive documentation of the students' problem-solving processes and actions was attained. 53 3.5.3 Artifacts To supplement the data collected through observations and recordings, I gathered student artifacts. These included drawings and written solutions some students had made on paper, and videotaped recorded images of construction of robots, which I then excerpted through screenshots. Further, I mined coding scripts from files that had been saved on the laptop. Students were using a software interface that employed a block-based programming approach. This method involves students manipulating and arranging different blocks (Figure 3.5) from a palette to create a sequence of actions or instructions. The names of these blocks are: (1) Start, (2) Motor This Way, (3) Motor That Way, (4) Motor Power, (5) Motor On For, (6) Motor Off, (7) Motion Sensing, (8) Tilt Sensing, (9) Text Input, (10) Number Input, (11) Random Input, (12) Display Input, (13) Play Sound, (14) Wait For, (15) Repeat. By dragging and dropping these blocks, students were able to build and structure a script or program in a visual and interactive manner. The code scripts produced by the students were stored on a laptop. Throughout the study, the pairs of students produced one (co-constructed) script rather than individual scripts; thus, each pair and each single participant produced six scripts, for a total of 24 scripts in all. Figure 3.5 A Screenshot of the Programming Block Palette 54 Figure 3.6 exemplifies one of the student's script sequences for the Clock/Windmill task. The code commands a motor to spin for a specified duration. It comprises four blocks executed within a loop: a Start block (1) for program initialization, an Output block (2) to dictate the motor's clockwise direction, a Wait block (3) to introduce a pause in the code (here, causing the motor to spin for 1 second, where a wait time of 10 represents 1 second), and a Repeat block (4) to iterate the entire operation 10 times. As a result, the windmill blades underwent a clockwise spinning motion for a duration of 1 second and repeated this action a total of 10 times within the loop. The collection of artifacts (written work/drawing samples, images of robot constructions, and coding scripts) provided additional evidence from which to determine students' problem-solving processes and their use of mathematics concepts. 3.5.4 Data Analysis Throughout the study, I observed the students’ participation in individual or paired work on pre-designed robotics activities, resulting in a total of 24 episodes with genuine contexts. The extended timeline of the study provided a unique opportunity to explore theoretical reasoning, Figure 3.6 A sample of Coding Script 55 including constructionism and SRL, through their direct application to practical, real-life scenarios. Accordingly, I analyzed data collected from various sources to describe and explore how students used self-regulated learning (SRL) strategies and mathematical knowledge and skills during robotics problem-solving activities. The initial analysis involved the following steps: 1. I transcribed video recall conversations, then segmented and classified them into aggregations that exemplified the students' use of mathematics and self-regulation. This process allowed for the identification of specific instances where mathematics and self-regulation were present in the students' verbal descriptions of their activities during the interviews. 2. To ensure systematic analysis, I developed separate coding schemes (Appendix B) for self-regulated learning (SRL) and mathematics, drawing from the categories established in Perry’s SRL study (1998) and Schoenfeld’s math problem-solving work (2010, 2011), respectively. I took a deductive approach to create these codes which directly aligned with the components outlined in those theories. These coding schemes were designed to ensure a balanced representation of indicators for self-regulation and mathematical thinking. 3. I meticulously examined the observations and field notes (an example of which is provided in Appendix C). During the process of labeling field notes, I applied the codes from Appendix B in situ, meaning that I assigned them as I directly observed the students. For instance, if a student remarked, "that is better," I categorized it as monitoring in self-regulated learning (SRL). Similarly, when I witnessed students 56 attempting to solve 13+3 either on paper or mentally, I classified it as the use of computation skills in mathematics. This approach enabled immediate and contextualized coding, capturing the subtleties and dynamics of the students' behavior, interactions, and problem-solving strategies in real-time. The concluding phase of the analysis employed a multi-tiered approach. It began with an in-depth examination of individual student behaviors and then expanded to encompass a collective analysis of students’ problem-solving processes involving all participants. I incorporated phases drawn from established problem-solving models and approaches discussed in the earlier literature review. Key frameworks included Polya's mathematical problem-solving model (Polya, 1971), the social self-regulation learning model (Zimmerman, 2000), engineering design thinking model (Mangold, 2013), and computational thinking (Wing, 2008). Drawing upon my understanding of these problem-solving models, I synthesized and reorganized the identified processes into phases to best represent the recurring patterns observed in the behaviors of young students in the context of robotics. Additionally, the final analysis process entailed examining how students identified and rectified errors stemming from mistakes, inaccuracies, faults, or unforeseen events (See an example in Appendix E). The focus on students' errors was intertwined with their assessment of progress, attitudes toward these errors, and the broader conversation surrounding perseverance in their actions. Errors in this context pertain to deviations or discrepancies between the anticipated outcome and the observed robot behavior. The initial examination of students’ errors encompassed a thorough assessment of the overall functionality of their robotic creations. The 57 process involved categorizing errors into major distinct types, i.e., coding errors and mechanical errors. A qualitative analysis was employed to understand students’ engagement related to the identified errors. This included reviewing interviews and field notes from the subcases, where students discussed their decision-making processes, problem-solving strategies, and the errors and setbacks encountered. Additionally, a comparative analysis was conducted to discern patterns across subcases and tasks, considering factors such as prior experience with robotics and individual problem-solving approaches. 58 Chapter 4: Results In this chapter I report results from a case study of six grade two and three students who were attending three learning centers where they used LEGO parts to build simple robots and code them to complete tasks. I explored two questions: 1. How did young children use mathematics as they engaged in robotics activities? 2. What self-regulated learning opportunities did robotics afford as young children engaged in problem solving activities? I observed some patterns in students' participation in robotics activities. These patterns emerged from a diverse set of qualitative data sources, which encompassed classroom observations, field notes, students' construction, and programming projects, as well as interviews. Observed across the activities of all six children throughout the study, these patterns were particularly noticeable in how young students: (1) participated in problem-solving. (2) engaged in decision-making and goal-setting. (3) perceived and applied mathematics in robotics activities; and (4) interpreted and responded to errors in their endeavors. 4.1 Iterative Problem-Solving in Robotics All participants in my study followed their own schedules as indicated in Table 3.1.2. The robotics program's curriculum allowed for flexibility, enabling students to choose their 59 participation in specific projects based on their interests and preferences. Consequently, students participated in different projects, highlighting the individualized nature of the curriculum and contributing to the diverse experiences of the students throughout the study. To provide the reader with an immersive understanding of the study's context and to demonstrate the iterative nature of the problem-solving of young children within robotics, I offer detailed descriptions of the Windmill/Clock activity as it unfolded for each participant. It serves both as an illustrative example of students’ engagement and actions in robotics activity more generally, as well as evidence supporting the iterative nature in which these children approach problem solving. 4.1.1 Individual Problem Solver in Action In the Windmill/Clock activity, students were assigned the challenge of creating a fully operational windmill or remodeling it as a functional clock. The instructors at Center B and Center C acknowledged that while the specifics of each option presented to the students may have varied, the core learning outcome for the windmill task remained consistent: to engage students in designing and coding. The instructor at Center C further commented that regardless of the chosen option, the task aimed to develop students' skills in modifying the windmill, programming the motor for specific speeds and directions of rotation, and fostering their understanding of how coding and customization could impact a structure's functionality. These instructors did not perceive the activity as primarily math-centric. Instead, the emphasis was on engineering construction and the practical application of coding. 60 Students were instructed by instructors to start the project by clicking the first button beneath the image of the activity. It was linked to the initial assembly of a vertical stand using Lego pieces. This task required them to use standard LEGO blocks along with some specialized pieces. The activity began with the assembly of the windmill's base, as illustrated in Figure 4.1.1. This base acted as a vertical support structure for the rotor blades and motor, following on-screen instructions. Once the windmill construction was complete, students accompanied me to an adjacent room, away from the robotics instructor, to engage in further problem-solving that was associated with the second button under the image. When the second button was clicked, the students were presented with two challenge options from which they could choose. One option prompted them to “make the windmill spin 1 round in 1 second.” With this option, students would find themselves grappling with the interplay between the power and speed of a motor. The other option presented the student with a visual aid—an illustration of a tower clock—accompanied by a concise direction to “draw, cut and put a clock (dial) on the mill. You can remove or add some LEGO pieces.” The description for the activity read, “Code the clock hand to move 1 round and then stop.” This option involved reimagining the windmill as a functional clock (See appendix A). Either option demanded a meticulous balance, as it required a specific Figure 4.1.1 The Assembly Windmill Base 61 number of rotations within a designated time frame. While not explicitly stated during the activity, this task inherently encouraged the application of mathematical principles rooted in the realm of time and angles (rotation). Next, I will provide detailed descriptions of the four subcases’ Windmill/Clock activity to offer readers access to these observations. Subcase 1: Sylvia Sylvia was the only participant in the third grade. Similar to other students, she initialized the windmill/clock project by reading the problem statement. She read it aloud word by word. “That is easy. I can make it spin more than one round,” she commented. She started to draw a clock dial on a piece of paper. After completing the drawing, she juxtaposed the clock face with that of the windmill stand, determining the dimensions of the mill’s stand. Recognizing the need to change the height of the stand, she embarked on the modification of the structure. She dedicated efforts to improving the windmill's stability, experimenting to ensure it would not easily tip over. After connecting the motor to the top of the mill, she realized she needed to make a hole in the clock face so that an axle could go through the paper clock face. She had to take the motor down to solve the problem. With a prototype assembled, Sylvia went on testing the creation. She focused on testing its robustness. A challenge emerged: the motor unexpectedly stopped spinning the clock hand. Upon careful investigation, Sylvia discerned that the issue lay with a loose axle which failed to firmly hold the paper clock Figure 4.1.2 Sylvia’s Clock 62 face in place. This led her to engage in a series of iterative experiments to find better ways of securing the paper in place. Eventually, she discovered a solution to the problem by adding a yellow LEGO piece to the axle to hold the paper in place (as shown in Figure 4.1.2.). The yellow piece was able to hold the paper against the stand securely, preventing it from slipping or becoming misaligned during spinning. The addition of this piece proved to be a simple yet effective solution to the problem, and she was happy she had successfully resolved the issue with her idea. She expressed her satisfaction with the building by saying, “Ok, now.” She quickly glimpsed the challenge description on the computer screen, “I have to code it . . . to move it around.” In her coding exploration, Sylvia took a straightforward approach. She began by selecting a yellow starting block followed by a motor spin block. This short sequence highlighted her intention to focus on the initial step of the task—getting the motor to spin—before getting into complexities like speed and precision. Sylvia also used this simple script to check if the computer and robot were communicating effectively, ensuring there were no connectivity issues between them. The test also helped her to evaluate how much of her goal had been achieved. With a sense of success with the first step, Sylvia exclaimed, “All right!” She proceeded to enhance her code by adding more blocks. She introduced a backward motor movement block which would reverse the motor. Such code would ignore the first movement and execute the second movement. She tested it and was happy with the smooth spinning, but she was surprised 63 by the speed of the spin. “Oh, my gosh, the clock is crazy!” she exclaimed as the hand moved very fast. She then began to experiment with different numerical input in order to reach a solution to make the clock spin only one round. Figure 4.13 illustrates the development of her coding endeavor. She started by inputting 5 to the timer. She ran the code and observed the clock. “Still too short,“ she murmured and changed the number to 10. The spinning cycle of the clock hand was shorter than she expected. Sylvia tested different inputs repeatedly. She continually fine-tuned the timing until number 34. Sylvia expressed uncertainty, stating, “I think...I don't know.” While she acknowledged that the solution was acceptable, she recognized its imperfection. I informed her that class was about to be over in a few minutes, she decided to try something to see if it would work better. “I'll go back to the first,” she said. She started to revert to the initial code by removing the time control and ran the code. It appeared that she was happier with the outcome. Figure 4.1.3 Development of Sylvia’s Code 64 Subcase 2: Tania Tania, a second-grade student, had an energetic demeanor. She was always enthusiastic about new activities that made her a dynamic participant in the study. In the windmill/clock project, she initialized problem-solving by carefully reading and studying the problem description and image on the computer screen. After reading the prompt on her screen, she murmured, “How do I do this?” She revisited the computer image of the clock tower. With that initial understanding, she started to cut circles out of paper and drew lines on the clock face (as shown in Figure 4.1.4a). She compared the finished clock face to the stand, and she decided to adjust the height of the stand to fit with the clock face. Then she confronted the challenge of securing the clock face to the stand. She experimented with different pieces and finally improvised to use tape to secure the pieces together (Figure 4.1.4b). Not having double-sided tape readily available, she crafted her own double-sided tape by combining single-sided tape pieces. This solution allowed her to successfully affix the clock face to the stand. Figure 4.1.4b Tania’s Clock Figure 4.1.4a Tania’s Clock Face 65 After resolving the mechanical issue, Tania went back to the computer and re-read the problem description and decided to move on to coding the robot. Her coding idea was to simply run the motor as shown (Figure 4.1.4.c). The clock seemed to be spinning but not as she expected. The clock face wobbled and oscillated due to being off-center. Tania had to re-engage in mechanical troubleshooting. She tried to reinforce the structure's foundation with additional pins, but it did not immediately solve the problem. She continued to experiment with the same idea, in the hope of improving its balance. However, the class concluded before she was able to achieve the desired result. Tania demonstrated an iterative problem-solving approach that was not explicitly guided but seemed to be a natural and integral part of her problem-solving process. Tania demonstrated a high level of engagement and eagerness during the “Initial Assembly” phase, showcasing her capacity for self-regulation. Her curiosity and enthusiasm were evident as she explored the task independently. However, as she transitioned to the “Challenge” phase, Tania encountered moments of hesitation when facing obstacles. During the challenge, Tania did not seek assistance from the instructor, showcasing a form of independence and persistence. Subcase 3: Carl and Princeton Carl and Princeton were both in the second grade. They took on the option of constructing a controllable windmill. Their collaborative effort began with quick online research into various Figure 4.1.4c Tania’s Clock Code 66 windmill blade designs, research that enabled them to collect ideas and insights. As they proceeded, Princeton sought to optimize the blade design. He extended the windmill stand to provide ample space for the four-blade design. During the coding stage, Carl and Princeton made use of a coding block (located in the middle and featuring a gauge icon in Figure 4.1.5), an approach not attempted by any other students in this activity. They independently chose to incorporate this block without any prompting from me. This new block provided them with the capability to regulate the power output, which could not be achievable using the common motor block. Through systematic testing, they examined all power outputs ranging from 0 to 10, ultimately determining 1 (equivalent to 10% of the motor's maximum power) to be the optimal choice. Their continuous assessment led them to conclude that this selection yielded the most favorable outcome. Their joyful comment — “It works!”— underlined their sense of achievement in the project. Princeton’s joy in the building process and Carl’s dedication and attention to detail supported the mutual determined approach towards the robotics challenges. Figure 4.1.5 Carl and Princeton’s Windmill and Coding Script 67 Subcase 4: Jason and Kent Jason and Kent were another pair in the second grade. Despite their busy after-school schedule, both actively participated in robotics activities. Opting for the windmill challenge, they set out to construct a windmill that could achieve 10 rotations within 10 seconds. Their initial assessment labeled the task as “easy.” They quickly completed the blades using a round brick and a 90-degree angle connector. They were excited to experiment with different speeds to see how the windmill would respond but their perception of the complexity of the problem soon shifted. After re-reading the problem description, they realized the primary goal was not to make the blade spin fast or slow, but to attain a specific speed. This prompted a shift in their strategy. Kent embarked on manual experimentation by adjusting the axle bushings and ensuring robust connectivity between elements, while Jason continued to refine the coding. Challenges surfaced as Jason and Kent grappled with the non-linear relationship between power input and speed output. The complexity of controlling the motor led them through an iterative process of experimentation with different code combinations. While Jason ran the code and observed the windmill spinning, he murmured, “1 Mississippi, 2 Mississippi . . ..” to help him track the time. During code testing, the motor's inability to slow down seems to pose a perplexing challenge to the team. Kent suspected the issue lay with the weight of the blades. Kent and Jason removed several pieces to lighten the blade. However, they realized that heavier blades yielded better results. After a series of failures to attain the desired speed, they altered their objective to a simpler one: making a code to command the windmill spin for 10 seconds while setting aside the 68 required rotation (As shown in Figure 4.1.6, which they had removed the speed control block). Navigating toward this more manageable task, they went on exploring varying numbers and configurations. At the end of the class, they managed to achieve a 10-second spin of the blade and they decided to incorporate the speed control back to it. Upon informing them that the class time had concluded, I noticed their reluctance to stop. Kent’s expression was tinted with disappointment, showing a desire to continue to improve their work. Jason made a sound, a heavy sigh that echoed the unwillingness to give up. 4.1.2 Cross-Case Problem Solving Approach: The Iterative Process The previously detailed depiction of each subcase’s problem solving during the Windmill/Clock activity highlights diverse approaches within a singular robotics activity. Yet, despite the variations observed across activities and subcases, problem solving during robotics activity involved four phases: understanding, ideation, experimentation, and assessment. The “understanding” phase involves grasping the foundational principles and specifications of the robotics task, including identifying key components and their functions. “Ideation” encompasses Figure 4.1.6 Jason and Kent’s Windmill and Coding script 69 the creative generation of potential solutions or approaches, translating theoretical understanding into practical ideas for constructing or programming the robot. The “experimentation” phase involves the hands-on implementation of the chosen approach, during which students actively built, programmed, and refined their robotic creations. Finally, the “assessment” phase centered on evaluating the effectiveness of the solution, considering factors like functionality, efficiency, and alignment with the initially identified problem or challenge. In Table 4.1, I provide examples of the types of evidence I observed for each of these phases as children completed their Challenges. Phase Examples Understanding Identifying the constructing parts such as motors, sensors, and connectors Reading through the task instructions Clarifying the task with a partner or instructor Discussing the key components with a partner or an instructor Ideation Brainstorming various ways to address the given challenge. Suggesting multiple ideas for constructing a robot or programming a robot Prioritizing the identified issues Considering factors such as stability, mobility, and sensor usage Discussing ideas with the partner Seeking feedback from the instructor before deciding Experimentation Assembling the robot according to a chosen design Programming the robot according to a chosen idea Testing different configurations of the robotic components to optimize performance. Experimenting with different sequences and variables to ensure the robot behaves as intended Assessment Observing the robot's movements and comparing to goals Evaluating robot’s ability to maneuver Questioning results Assessing robot’s overall performance Identifying areas for improvement, such as adjusting motor speeds or refining the code Table 4.1 Problem-Solving Phases and Examples 70 Even though all cases had distinct inclinations and problem-solving strategies, the problem-solving processes of each subcase (a student, or a pair of students) unfolded in similar ways, intertwining the phases of understanding, ideation, experimentation, and assessment, often cycling through the latter three phases multiple times. Each iteration observed in this study consistently commenced with the understanding phase, during which students absorbed the problem’s nuances and engaged in endeavors to understand the expectation and boundaries. Students achieved this by reading the problem description, examining the provided images, seeking clarifications from the instructor, and discussing the task with their partner. Then they transitioned into the ideation phase, to brainstorm ideas. Transitioning into the ideation phase did not always involve vocal indicators. Some of the signals indicating a quiet shift into ideation included spending more time quietly reflecting on the assessment; appearing contemplative; rearranging their experiment workspace. As students transitioned to the experimentation phase, they transformed their ideas into tangible objects, testing their feasibility through hands-on experimentation. This phase involved constructing, refining, and adjusting their robot designs and coding scripts. This phase naturally led them to, or overlapped with, the assessment phase in which they evaluated their robots’ performance and the effectiveness of their coding strategies. This evaluation served as the foundation for their next cycle with each iteration guided by insights gained from the previous cycle. This simple yet flexible loop (Figure 4.17a) allowed 71 students to build upon earlier ideas and insights, ensuring incremental improvement towards the final solution. In Figure 4.1.7b, I provide a visual representation of the iterative problem-solving approach observed in the Challenge stage of the Windmill/Clock project for each subcase, which I have previously described in detail. In Figure 4.1.7b, as in Figure 4.1.7a, each distinct phase identified in the iterative problem-solving process is represented by circles which are color coded (i.e., understanding (blue), ideation (green), experimentation (red), and assessment (yellow). The illustration in Figure 4.1.7b is structured into two dimensions: building (e.g., stabilizing or extending the windmill base) and coding (e.g., assembling coding blocks to achieve functional requirements). Each dimension represents a significant endeavor, depicting the nature of major efforts involved in constructing the robot and programming it. These two dimensions are presented spatially (Figure 4.1.7b), with iterations of building situated below the horizontal axis and coding above it. The horizontal axis itself (blue arrow, Figure 4.1.7b) signifies the timeline from the beginning to the end of the problem-solving session, with the left-to-right positioning of the circles depicting the sequence in which these actions occurred. Time lapsed was not analyzed Figure 4.1.7a Iconic Representation of Understanding, Ideation, Experimentation, & Assessment Cycle 72 and is not represented in this diagram. The illustration depicts a trajectory, with the challenge of capturing the dynamic interplay and intertwining of the phases in real-time. Students initiated their problem-solving process by first engaging with the presented problem (i.e., labeled as understanding the problem and represented as a blue circle shape). From there, they deliberately entered either the construction or coding stage. During periods of building and/or coding, each subcase was seen participating in iterative processes of ideation, experimentation, and assessment (represented as a triad of different colored circles with arrows indicating a clockwise movement from one to the other). As can be seen in Figure 4.1.7b, their individual experiences diverged as they engaged with this problem-solving approach. Figure 4.1.7b Each Subcase’s Iterative Problem Solving During Challenge: Windmill/Clock 73 Throughout the Clock/windmill activity, it can be seen that all students initiated their iterations with the building stage, even though there were no explicit instructions guiding them to do so. Not all iterations adhered to a fixed sequence. Students intentionally charted diverse paths, branching into building and coding stages. Three of the four subcases chose to revisit the posed problem (shown as understanding (blue circle)) at one other point within their problem-solving process. Understanding the problem, therefore, did not solely occur at the outset of the problem-solving processes of the illustrated example (Windmill/clock). That said, Carl and Princeton’s case did not include additional understanding within the duration of their activity (Figure 4.1.7b, Carl & Princeton). It is unclear from the data whether this was because their initial understanding was sufficiently robust that adjustments were not needed, or if time committed to the iterative processes of ideation, experimentation, and assessment, which in turn resulted in a successful solution, lessened the need to reflect further on their initial understanding. Regardless, what is noteworthy here is that this dynamic and cyclical process of understanding, ideation, experimentation, and assessment contributed to the refinement of a solution to the problem, where each iterative cycle acted as a foundational anchor for the subsequent one. To further illustrate these four phases of the iterative process, I provide a more detailed description of one subcase. To do so, I reframe my earlier description of Sylvia’s (Subcase 1) clock activity (see Table 4.2) in a succinct sequential manner to more readily foreground the iterative process in action. Sequence Phases Observations/Transcripts 1 Understanding Sylvia read the problem statement aloud. She asked for crayons and paper to draw clock face. 74 Sequence Phases Observations/Transcripts 2 Building Ideation She observed the dimensions of the stand and identified the issue of stability. 3 Experimentation She worked on the structural modifications, ensuring the stand's height met the desired criteria. 4 Assessment Sylvia examined how the clock face was attached to the stand and recognized the need for something more secure. 5 Building Ideation She looked closely at the center of the clock face, engaging in deep thought. 6 Experimentation She worked on creating the hole in the clock face and simultaneously experimented with motor placements. 7 Assessment She manually tested the spin, and she observed the entire structure from different angles. Sylvia said “Ok, now…”, indicating her satisfactory evaluation of the construction. 8 Understanding Sylvia revisited the problem description on the computer screen and confirmed that a key component of the challenge involves “move it around.” 9 Coding Ideation She initiated her coding exploration by testing communication and connection to the robot. 10 Experimentation Sylvia started experimenting with one block and expanded the program to include more blocks; she tested time control from 3 to 34. 75 Sequence Phases Observations/Transcripts 11 Assessment She was surprised by the speed of the spin, saying “crazy”. I suggested a break, but she insisted on trying a new idea. 12 Coding Ideation She focused on addressing the identified issues, which were the overspeed and lack of stability. Her main priority appeared to be reducing the speed to avoid any undesirable outcomes, which she referred to as "crazy." 13 Experimentation She took action by using the “undo” button to reverse the code. Pausing momentarily, she carefully checked each undo step. 14 Assessment She said “ok, ok” with a cheerful tone. I interpreted this as she thought the previous solution was superior, showcasing her contentment with the overall result. Table 4.2 Problem-Solving Iteration Example Remarkably, this dynamic and iterative problem-solving approach persisted among all students across various activities. Whether they were working on a windmill, designing a carousel, or making a seesaw, the students consistently applied this iterative process. They tackled smaller challenges as steps toward solving the broader problem. This shared approach indicates that they had embraced this dynamic and iterative problem-solving method as a valuable and effective approach in the realm of robotics tasks. However, there were cases, such as Subcase 2 (Tania), where her involvement in the Walker project presented a challenge in understanding a text description to construct a ramp for a robot to climb over. Tania struggled to comprehend the 76 challenge, and as a result, her engagement in the iteration process was very limited. Instead of participating in the problem-solving iterations, she shifted focus on fixing the initial assembly and spent the rest of class time creating a spider game using LEGO pieces. Understanding the challenge played a crucial role, acting either as a hindrance or a catalyst in progressing to subsequent stages of problem-solving. 4.1.3 Strategies for Problem-Solving in Robotics While I previously described the children’s problem-solving approach, I focus here on the problem-solving strategies students exhibited as they engaged with the robotics challenges. In particular, decomposing, trial-and-error, and self-assessing were strategies which appeared to play an important role in shaping the problem-solving process of the students in this study. Decomposing. The process of decomposing problems into smaller, manageable tasks was a common skill that students used during what I previously labelled as understanding the problem as posed. Students broke down the posed problems into smaller steps, either in parallel or sequentially. The following examples illustrate some of the ways individuals or pairs used decomposition: See-saw/Hammer. Build and code a seesaw that goes up and down or a hammer that hits a nail. Tania (See-saw) Carl and Princeton (See-saw) Sylvia (Hammer) Session 3, Center A Session 2, Center B Session 6, Center C Figure 4.1.8 See-saw/Hammer Excerpt from Table 3.2 77 The see-saw and hammer activity (see Figure 4.1.8) involved an interchangeable base structure. By modifying the pivot point of the base structure, the functionality of the object transformed from a seesaw to a hammer. The see-saw consisted of a beam with a blue seat and a red seat placed at opposite ends. In the see-saw project, students were tasked with getting 26 points using their robots. When the red seat was up, it earned 3 points, while the blue seat earned 5 points. The hammer project, on the other hand, introduced a different scoring system. Here, students aimed to earn 26 points by performing different types of hits with their robots. A “smash” was worth 5 points while a “soft hit” scored 3 points. The coding challenge in this project involved programming the robot to execute the appropriate type of hit to reach the target score. Sylvia employed a partitioning strategy, dividing the problem into two parallel subproblems: adding 5 and adding 3 respectively. Each subproblem could be solved independently without relying on the success of the other, but both contributed to the final sum. Figure 4.1.9 Sample of Sylvia’s Decomposition 78 By treating the addition of 5 and the addition of 3 as separate subproblems (Figure 4.1.9), Sylvia was able to address each one individually. This approach allowed her to solve the subproblems in parallel, independently of each other. Figure 4.1.10 Frog Excerpt from Table 3.2 In the frog activity (see Figure 4.1.10), Subcase 2 of Tania chose to feed the frog using three small meals she constructed from LEGO blocks. While working with her robot, Tania showcased a systematic decomposition approach. She divided the problem into three sequential actions: activating a motor to open the frog's mouth, producing a sound, and waiting for a specific duration for the next round. Tania diligently focused on each step, ensuring success before moving on to the next component. Ultimately, she repeated the successful sequence three times in her final code (as shown in Figure 4.1.11). Her approach exemplified the method of breaking Frog Build a frog that makes an interesting sound when it eats. Tania Session 6, Center A Figure 4.1.11 Sample of Tania’s Decomposition 79 down the initial problem into manageable components: addressing the first step of the sequence first, advancing through the subsequent individual components, and ultimately tackling the original challenge. Bulldozer Build and code a bulldozer capable of picking up LEGO blocks from a pile. Jason and Kent Carl and Princeton Sylvia Session 6, Center A Session 6, Center B Session 2, Center C Figure 4.1.12 Bulldozer excerpt from Table 3.2 In the Bulldozer activity, Carl and Princeton took on the challenge of designing a robot capable of “scooping up” LEGO bricks (Figure 4.1.12). Rather than directly attempting the final solution, Princeton introduced a step-by-step approach by suggesting the robot “turn that way and stop in the middle”; he focused on the partial action of “scooping” rather than the complete cycle. After successfully completing the coding of pushing, Carl and Princeton enhanced their creation by stabilizing a mechanism capable of both lifting and scooping the bricks. This approach allowed them to gradually improve the robot's functionality and move closer to resolving the initial challenge. Jason and Kent in another session showcased a different but similar strategy. Jason initiated the process by concentrating on pushing the bricks. It was evident that the pair recognized pushing as a preliminary stepstone, not the final solution; they continued working on additional movement of the robot after they successfully achieved pushing. The above examples showcase the diverse approaches students took in decomposing complex 80 problems into more manageable tasks. By simplifying the initial problem into smaller manageable elements, and then progressively introducing complexity, the students reduced the overall complexity of the initial problem. This approach allowed them to steadily advance toward achieving the final solution. Trial-and-Error. Trial-and-error was a common problem-solving approach used in the experimenting phase, where students troubleshot the issue using different configurations in construction, or different sequences of coding in programming. Here are some examples: Figure 4.1.13a Carousel Excerpt from Table 3.2 Jason and Kent were involved in building and coding a rotating carousel that stops at specific intervals (Figure 4.1.13a). They employed trial and error to determine the appropriate timing and motor control to achieve the desired stopping intervals. Their approach primarily relied on trial Carousel Build and code a four-seat carousel to spin at an interval period of time. Jason and Kent Session 3, Center A Figure 4.1.13b Trial and Error: Jason and Kent’s Carousel Coding Script 81 and error to determine the precise timing and motor control necessary to achieve the desired stopping intervals. As illustrated in Figure 4.1.13b, Jason and Kent initiated their experimentation by testing various stop times, beginning with 19 and progressing to 52, and ultimately reaching number 204. This method allowed for a structured exploration of potential solutions with the flexibility to refine and adjust the variables that could be changed while keeping the known parameters unchanged. With each trial and each adjustment, Jason and Kent conducted swift assessments of the performance of the robot which showed they were sometimes closer to the goal and sometimes further away. Their pursuit led to the successful timed stops. Figure 4.1.14 Walker Excerpt from Table 3.2 During the Walker project, Tania encountered a challenging situation at the project's outset (Figure 4.1.14). Her assigned task was to program a walker robot to initiate movement from one point and come to a stop at another. Tania incorporated numerous trial-and-error attempts into her code, each representing a different guess. Unfortunately, despite her efforts, the robot consistently fell short of covering the intended distance. She reviewed her code, made adjustments by adding another motor block, and continued experimenting. In this particular Walker Build and code a robot to climb up a ramp or go a certain distance. Carl and Princeton Tania Jason and Kent Session 4, Center B Session 4, Center A Session 1, Center A 82 example, Tania relied on a trial-and-error approach, making multiple guesses. While this method did not yield the desired outcome within the allotted time frame, Tania's dedication was evident in how she persistently explored various adjustments and modifications in pursuit of improving the walker's movement. I have described two trial-and-error approaches, systematic and random methods, which were also observed combined or blended in the students’ work. The above examples underscore the importance of trial and error within the students' problem-solving endeavors in robotics. By repeatedly cycling through the process of experimenting with various ideas, closely observing the robot’s performance, and then making necessary adjustments, students enhanced their creations and achieved improved outcomes. Self-Assessing. The robotics activities provided students with non-routine challenges and offered concise descriptions of the desired outcomes. This clarity in instruction played a significant role in shaping the iterative paths of the students. However, there remained flexibility for students to adapt and interpret these instructions in their own ways. Notably, there were not rigid criteria beyond the overarching goal, allowing for a more exploratory approach and self-assessment of the tasks at hand. Students made effective use of real-time feedback from the robots, and feedback from their peers (if working in pairs), to assess their progress, identify areas for potential improvement, and make immediate refinements to their solutions. The diverse methods these children employed for self-assessment will be elaborated upon in the following sections. 83 Co-constructing criteria for evaluation played a role in the windmill activity, where Subcase 3 (Carl and Princeton) demonstrated their assessment of the performance of the robot. Carl: Look, it is too fast. Princeton: I did not see. Redo it. Carl: [re-run the code] Princeton: Yeah, too fast. The challenge description provided no specific information regarding the required speed of the blade. The conversation between Carl and Princeton shed light on their engagement in assessing their robot's capacity to meet a speed criterion they had established themselves. Their evaluation primarily centered on comparing the robot’s performance with the predefined speed goal they had set. Carl and Princeton's approach illustrates their involvement in defining performance standards, a practice that was found across various activities in this study. Subcase 4 (Jason and Kent) in the Windmill/Clock project provides another example of shared assessment. Initially, the pair shared an optimistic perception of the challenge. Jason said “easy” and Kent nodded. The pair collectively assembled the blades, showcasing a joint understanding of the initial task. However, their understanding of the task evolved when Jason re-read the task description and realized that the two factors needed to be simultaneously considered: one round of spin, and one second. This concept of speed, seemed novel to both students. The discussion between the two students led to a decision to prioritize the number of the spin over the time of the spin. Their collective decision to alter the objective and focus on a simpler task showcased shared regulation in the assessment phase. 84 The next example showcases another positive collaborative aspect of learning through joint evaluation between Jason and Kent. In this instance, they cross-checked each other's math work to ensure accuracy. They were asked to find out the total amount of the studs on a base plate, which is comprised of 16 rows, each housing 8 studs (Figure 4.1.15, left). Kent initiated the task by counting the studs one by one using his finger. Jason took a different approach by employing a vertical multiplication method on a piece of paper. Following this, Kent abandoned an incomplete counting attempt and took the initiative to double-check Jason's work after Jason stated that he had obtained the number. Kent recorded his calculations beneath Jason’s work using the same method. There was a minor discrepancy between their methods: Kent's carry-over Figure 4.1.15 Cross-Checking Calculation 85 of 4 landed at the bottom of the number, while Jason’s carry-over placed the number 1 on the shoulder (Figure 4.1.15, right). Both students smiled at each other when they discovered that they had arrived at the same result, a total number of 128. This interaction seemed beneficial for completing the task: Jason had the opportunity to have his work reviewed, while Kent contributed to problem-solving by identifying errors or oversights. The act of joint evaluation demonstrated a sense of teamwork and mutual support throughout the process. As part of self-assessing, students identified both successes and failures. As an example of identifying success, Sylvia planned to code the clock to spin more slowly. She simply looked at the clock upon each new input number on her computer. The instant response from the robots enabled her to make real-time adjustments to her solutions. When she noticed the speed was too high, she quickly identified the coding block and modified the number. She kept observing the impact of her adjustments on the robot's behavior and performance until she was satisfied with the outcome. “That is better,” she said. Acknowledging her success, no matter how small, seemed to help her maintain motivation and a positive attitude. Identifying failure was another aspect of self-assessment. Jason and Kent were working on a robot that could move in a walking fashion. When they noticed that the motor was spinning but the robot did not move forward, they quickly realized that something was wrong. Jason said, “This is weird.” “We did something wrong,” Kent confirmed. Both students expressed their realization that they had made a mistake. Without seeking assistance or evaluation from me, they continued to make changes in their code. 86 Students also engaged in tracking progress on their own. For instance, Tania took the initiative to embark on the Tower/Crane project when she was informed that the instructor would be late for the class. The Tower project provided two options from which students could choose, either to build and code a tower that was able to reach a height or to build and code a crane. Tania carefully studied the options available and chose the option of making a tower. As Tania worked on her project, without the presence of the instructor, she made comments to herself such as “this looks fine” and “how about one more piece here?” It was apparent she was engaged in self-assessment as she evaluated her own progress and determined her level of achievement. It seemed evident that engaging in self-assessment empowered students to conduct analysis of their problem-solving approaches, explore alternative strategies, and consistently evaluate their progress. Students contextualized the assessment process, with a focus toward understanding and addressing challenges specific to the task at hand. The immediacy of feedback provided by the robots seemed to play a role in cultivating students' continuous self-assessment practices. Observing the robots' movements and responses enabled students to discern the effectiveness of their code or structures. This dynamic feedback loop, facilitated by the robots, allowed students to evaluate their solutions based on predetermined criteria or criteria they established themselves. Notably, the inclination to seek external evaluation, either from me or the instructor, was infrequent. Instead, across all subcases, students autonomously gauged the performance of their code or mechanical structure, independently or collaboratively. Overall, this study showed that participants’ approach to problem solving in robotics was iterative in nature, in that children’s understanding of the problem develops in phases punctuated 87 by experimentation with tangible objects. In turn within the context of robotics, children’s key problem-solving strategies included decomposition, trial and error, and self-assessment. These problem-solving strategies collectively contribute to the development of solutions, allowing students to navigate and respond to a variety of challenges in the after-school environment. 4.2 Choice Making and Goal-setting in Robotics This study revealed that robotics activities offered students opportunities to establish personalized learning goals within projects, enabling them to pursue areas of interest aligned with their individual passions and strengths. Students demonstrated a notable degree of autonomy in goal-setting and decision-making, not confined to strictly adhering to prescribed task descriptions on computer screens or conveyed by the instructor. An illustrative example of this adaptive and goal-oriented approach was observed in Subcase 1, in Sylvia's participation in the Butterfly project. Sylvia went beyond the prescribed guidelines to make “a flipping butterfly” by murmuring to herself - “a beautiful butterfly.” Sylvia's self-talk provided valuable insights into her thoughts, showcasing her aim to blend technical proficiency and artistic sensibility. This internal goal-setting underscored the depth of her engagement and her aspiration to bring the project a sense of creativity. Sylvia's participation in the Clock project serves as another example of this internal goal-oriented process. She said to herself, “I will need a base,” and went to work on the idea. It revealed her approach to problem-solving, emphasizing the necessity of a stable foundation before commencing the other aspects of the structure. Sylvia's decision-making process is noteworthy, as it provided an illustration of her approach to navigating the intricacies of the task at hand. It 88 showcases not only a systematic approach but also a skillful breakdown of the larger challenge into smaller, more manageable tasks. Moreover, Sylvia exhibits a capacity to prioritize these tasks and establish goals. The nature of these non-routine activities appears to have played a role in fostering Sylvia's autonomy and responsibility in decision-making and goal-setting. By engaging in tasks that lacked predefined steps, she not only enjoyed the freedom to explore, but also consistently engaged in decision-making. In Subcase 2, Tania showcased a personal connection to her goals, making them more meaningful to her. In the Walker project, Tania integrated storytelling elements to share her narratives that highlighted the significance of personal connection to the project. Figure 4.2.1 captured a moment in her project where the bubble function became a storytelling tool, allowing Tania to depict a scene from her imaginative narrative. This integration of storytelling not only added a layer of personal expression to the project, but also highlighted Tania's choice-making process. It appeared that both Sylvia and Tania seized the opportunity to align their project goals with their distinctive interests and aspirations. Figure 4.2.1 Tania’s Walker Code 89 It is noteworthy that in In the Frog project, Subcase 2 (Tania) set a goal that diverged from the predefined task. Rather than simply adhering to the pre-outlined requirement of making the frog eat three meals, Tania expressed her goal of feeding the frog with only two meals. Tania: This is the food for lunch. [Tania put all the yellow blocks in one pile] Me: Then where is the food for breakfast? Tania: He doesn’t eat breakfast. Me: He doesn't? Tania: Because he always eats a lot of food at lunch time like a giant. Tania demonstrated an approach that went beyond mere functional considerations regarding the robot. Tania's decision to have her frog skip breakfast, contrary to the predefined task, revealed a nuanced and socially meaningful aspect to her goal-setting. Her specific emphasis on the varying amounts of food consumption for each meal of the frog might be indicative of the influence of her family's table talk, or from reading material outside of this program. By incorporating a narrative around the frog's eating habits, Tania added a personalized and potentially culturally influenced dimension to the project. The subcases of the paired students Jason and Kent and Carl and Princeton unveiled the dynamics of collaborative decision-making. In the next excerpt, the robotics task in focus required Jason and Kent to create a ramp for a robot to traverse. 90 Jason: Um, [reading the instructions on the computer] use paper to build a strong ramp for the robot to drive over. Jason: We need paper [turning to me to get paper for him] Kent: We need a coin. [he saw in the illustration that the task involved using coins to test the strength of the paper ramp] Me: What do you need, paper, or coins? [I ask them] Jason: Look, it says paper ramp. [he pointed to the computer screen] Kent: Yeah, we need a coin later. This excerpt captures the collaborative decision-making process, showcasing how Jason and Kent approached the task with distinct focuses. Jason concentrated on the conceptualization and brainstorming aspects, envisioning how paper could form a functional ramp, while Kent directed attention to the experimentation stage, particularly concerning testing the ramp's strength using coins as he noticed in the illustration accompanying the description. In another example, Kent's involvement in the Windmill project showcased a collaborative decision-making process by inquiring of his partner, Jason, “How to make it spin faster?”. His inquiry implied his goal of achieving the desired rotational speed of the windmill, highlighting his dedication to working with his peer to optimize the windmill's rotational speed. The dynamics of decision-making were also evident in subcase 3, Carl and Princeton, as they discussed their Sensor Car project. 91 Me: What is the hardest part? Carl: It is when we were trying to make the marker touch the paper. Me: Did you get a plan for that? Carl: Yeah, I got a plan; but he [Princeton] got a different plan. Princeton: [grinning when I turned to him] Me: Did he get the same plan or a different plan? Carl: Different plan. Me: How do you know? Carl: Because he seemed to not like my plan or anything related to my plan. Princeton: [relaxed when listening to Carl] Carl initially proposed constructing a small case for the pen, while Princeton suggested using a rubber band to secure the pen onto the robotics arm. Despite Carl's initial inclination to stick with his plan, he eventually agreed to give Princeton's idea a try. The implementation of Princeton's plan proved successful. Reflecting in the interview, Carl acknowledged the divergence in their initial approaches. This situation underscored the team's diverse viewpoints, where members contributed distinct ideas and strategies to tackle challenges. The willingness to adapt and evolve strategies based on collective input showcased the team's commitment to continuous improvement and the dynamic nature of problem-solving in a collaborative setting. Collectively, these above examples highlighted the richness of decision-making within robotics activities, on their own or collaboratively. Their commitment to setting and achieving clear objectives underscored a profound sense of purpose and determination. In observing their 92 interactions, it became evident that these students were not merely participants but actively regulating their learning, by demonstrating the capacity to make decisions and showing an awareness of the significance of goal in driving their progress. 4.3 Applications and Perceptions of Mathematics in Robotics This study embraced a broader definition of mathematics. It encompassed not only specific mathematical concepts and skills but also acknowledged mathematics as a tool for reasoning and analyzing, communicating, and presenting. In this section, I aim to report not only the practical instances where mathematics came to life in the construction and programming of robots but also the perspectives that young students brought to this learning environment. 4.3.1 Students’ Applications of Mathematics in Robotics The construction of a robot necessitated working with physical objects (e.g., LEGO blocks, paper). This involved matching the visual characteristics of the representation on screen, such as holes, studs, angles, and shapes, to the corresponding physical objects. The children employed various strategies in selecting their building elements. For example, Carl counted holes of a beam, while Tania focused on aligning the physical piece with the existing visual representation to make sure they were the same length and shape. Sylvia employed a color-based approach, using the color scheme to locate specific pieces (e.g., a yellow brick indicating a 60-degree angle). Jason experimented by rotating gears to different positions until the angle allowed both teeth to engage properly. Other than the obvious engagement with geometrical properties afforded by the materials, 93 students applied many other mathematics concepts through their physical representations (built structures) and symbol representations (coding scripts). I begin by highlighting the unique contributions of a single subcase then move to a cross-case report. Sylvia's creation of a butterfly-like structure involved an exploration of, at the very least, the use of symmetry principles. Sylvia folded a piece of paper along the middle line, and drew one wing on one side of the fold, then mirrored it onto the other side of the paper, thereby representing her understanding of symmetry in a tangible and visual form. She cut out the wings and attached them along the center axle of the motor base, as depicted in Figure 4.3.1. In building the butterfly, Sylvia made sure to use identical pieces along the symmetrical axes (the red 7-hole beams on the sides of the motor). However, she encountered an issue with placing the motor in the center of the structure due to the limited space available. She then used an extra gear to connect the axle of the motor which would synchronize the spinning of both red beams. This ensured that the wings moved together in harmony, maintaining the symmetry of the butterfly structure while in motion. Figure 4.3.1 Sylvia’s Robotics Butterfly 94 Additionally, Sylvia incorporated recurring patterns in her code, further emphasizing the concept of symmetry through yet another representation. As shown in Figure 4.3.2, the code made equal spins in both directions, which moved wings evenly on both sides of the body. By incorporating symmetrical patterns and movements in her code, she also implemented symmetry within the instructions given to the robot. Thus, in this project, Sylvia represented symmetry visually, through movement, and in programming script. In constructing a windmill, Jason and Kent applied their knowledge of geometry and engineering principles. They had to consider factors such as the size and shape of the blades, the positioning of the windmill on its base, and the overall stability of the structure. The physical representation allowed them to visually grasp these concepts in a real-world context. They could see the blades rotating in a circle and feel the vibration as gears moved other components. In addition, Jason and Kent showcased their understanding of multiplication and division concepts through their coding representation. Their goal was to achieve a 10-second duration of movement for the windmill. To accomplish this, they used a loop block in their code, symbolizing a repetition of 10 iterations. By iterating the loop 10 times, they effectively achieved a duration of 10 seconds, Figure 4.3.2 Sylvia’s Butterfly Code 95 which can be viewed as an alternative representation of multiplication (10 x 10). As such, in this activity, Jason and Kent employed a kinetic representation of the windmill's movement, and a coding representation of the sequence of action. Thus, rather than representing a concept in multiple ways, this robotics experience led these children to engage with different representations of multiple concepts. Creating a robotics clock engaged Tania and Sylvia in multiple representations, particularly visual representation of mathematical concepts related to an analog clock and including numerical sequencing, geometric properties, and representation of continuous gradation of time. Drawing from their memory, both Sylvia and Tania created a clock face that resembled a realistic analog representation. Sylvia placed the numbers unevenly on the clock face (Figure 4.3.3, Left) and thus numbers 3 & 9 and 12 & 6 were seriously misaligned across the center. In contrast, Tania’s numbers were better aligned, and she also added short and long lines, indicating her awareness of the minute and second markings visible on some analog clocks. Figure 4.3.3 Clock Faces Made by Students (Left: Sylvia, right: Tania) 96 While coding her clock, Sylvia encountered an interesting observation: her clock hand was moving in the opposite direction compared to a regular wall clock. Recognizing this difference, she expressed her desire for the clock hand to move in a clockwise direction. To address this problem, she adjusted the coding of the motor block's direction. Sylvia introduced a second direction block to ensure the correct clockwise movement, as depicted in Figure 4.3.4. Sylvia paid special attention to ensuring that the robotic clock hand completed a full rotation, representing the passage of time on an analog clock. On the other hand, Tania's coding script was more concise than Sylvia's. Instead of employing a second moving block, Tania used one moving block, which performed effectively in the desired direction. Her primary emphasis appeared to be on developing a functional clock that operated continuously without interruption, as evident in her coding displayed in Figure 4.3.5. She did not include the number of rotations required to halt the clock hand. In both cases, representing an analog clock entailed a drawing of the face, a construction of the ‘hands,’ and the programming of the hands’ movement, all of which pointed to various mathematical properties of this time-measuring tool. Figure 4.3.4 Sylvia’s Clock Code Figure 4.3.5 Tania’s Clock Code 97 In working on different activity options (see-saw or hammer), Sylvia, Carl, and Princeton showcased their capacity to represent mathematical concepts through multiple forms in the context of robotics. The see-saw configuration encompassed a beam housing distinct blue and red seats situated at opposing ends of the beam, while the hammer variant necessitated an up-and-down hammer mechanism for striking nails. Common to both alternatives was the objective of achieving a cumulative score of 26 points via robotic actions, using unique scoring rules. Upon the successful construction of a functional robotic hammer (Figure 4.3.6), Sylvia's endeavors shifted towards accruing points. Initially, she took a symbolic approach in which she employed mathematical equations on paper to methodically dissect and address the problem. Sylvia created an equation to encapsulate the interrelation of seat positions and the corresponding point allocations (Figure 4.3.7). She explored different starting positions for the Figure 4.3.8 Sample of Coding Representation Figure 4.3.7 Sample of Symbolic Representation Figure 4.3.6 Sample of Physical Representation 98 seats, considering various possibilities. However, through her symbolic representations and analysis, she realized that her idea of starting from 3 points (the second line of Figure 4.3.7) was not feasible and abandoned it. Subsequently, Sylvia translated her equation into executable coding blocks, employing the addition block (as illustrated by the block in Figure 4.3.8 bearing the plus sign) to symbolize the summation of points to culminate the values 5 and 3 respectively. In contrast, Carl and Princeton used oral representations of the sum, relying on mental calculations. As the see-saw shifted positions, Carl spoke the sums out loud, “3, 6, 9, 12.” This indicated his attempt to keep track of the accumulating points. Their code was a simple motor movement that drove the see-saw, without integrating blocks to add or display sum on the screen. Their approach involved using the kinetic representation of the robot and mental calculations together to represent the total score. The examples above showcase a diverse array of ways in which mathematics was integrated and applied to robotics by students. These adaptations provided students with the opportunity to use mathematical concepts through various means that resonated with their individual preferences in problem-solving. These included employing a variety of formats such as symbolic representations, physical embodiments (constructed robots), dynamic representations (robots in motion), coding as a form of iconic representation, cognitive processes, and verbal articulation. 4.3.2 Students’ Perception of Mathematics in Robotics. The students tended to underestimate the role of mathematics in their problem-solving efforts and the contributions it made to their achievements. When asked what mathematics they used in 99 solving the problem, Princeton acknowledged using counting, while Jason and Kent recognized the application of multiplication and addition in the Tower/Crane project. Tania mentioned that she “counted” in the Butterfly project. During the Bulldozer activity, when I asked Carl if he wished to use more math, he expressed a disinterest in using math and specifically mentioned multiplication as an example of mathematics’ boring nature. Me: Do you wish to use more math in this activity? Carl: I don’t wanna use math. Me: Oh? Carl: Math is boring, and it is all 10 times 10, 10 times 9. (NB: implying times tables) In addition to signaling his negative perception of math, Carl’s response seemed to suggest he was unaware of using math during his problem-solving up to this point. Sylvia appeared confused or surprised by my question about using math in her Catapult activity. Me: What math did you use in this activity? Sylvia (shaking her head): “What? I don't know.” The fact that these students did not explicitly recognize the mathematical aspects inherent in robotics activities was not surprising, as the primary focus of the activity was to develop skills in building structures and coding. However, their responses revealed a limited perception of mathematics, often equating it with counting and basic arithmetic operations. Notably, there was a significant disconnect between students' perceptions of their use of mathematics in robotics and 100 the mathematics they actually applied. When their robots performed successfully, they attributed it to their coding skills, often responding with statements like “I coded it”. When asked about how mathematics contributed to their success, they seemed unaware that both their coding and building processes involved mathematics. 4.4 Recognition and Perceptions of Errors in Problem-Solving During the process of building and programming, students experimented with many alternatives and made many different choices about materials and codes to use. When these choices resulted in unexpected results or malfunction of the robots, I deemed that there were some types of setbacks and errors. Students viewed these as evidence of imperfection rather than failures. They took intentional steps to identify and isolate the specific errors, and subsequently, correct them. This positive mindset aligns with the principles of SRL, showcasing their self-monitoring and self-assessment skills. 4.4.1 Students’ Recognition of Errors while Building and Coding Robots While each student exhibited a degree of attentiveness during constructing the robot, the process of building the robot was not devoid of missteps. These errors included using the wrong piece, wrong size, wrong position, wrong orientation, and choosing the wrong port for electronic elements. Among these errors, the most frequently recognized mistake by students themselves was using the wrong piece. Using the wrong piece during construction often led to engineering issues and challenges. For example, using the wrong angle piece could cause the motor to stall, while using the wrong gear could result in unstable movement. Additionally, using the wrong-sized beam could obstruct the spinning elements or interfere with the intended functionality of 101 the structure. The emergence of these building errors often came to light when students put their code to the test or sought to execute particular movements. This dynamic feedback prompted them to confront and acknowledge their errors, compelling them to take remedial actions to adjust and rectify their architectural composition. Sometimes students did not explicitly articulate their errors in words, but their actions and immediate responses demonstrated their recognition of errors. For instance, Jason, who considered himself a “LEGO expert” and enjoyed building with LEGO at home, found himself making mistakes by using the wrong types of pieces in the Walker project. He used the wrong angle pieces, resulting in the legs of the Walker robot getting stuck. Although he did not verbalize his mistake, he immediately fixed the problem by changing the angle pieces, demonstrating his realization of the error. In programming their robots, all students experienced failures during the study. When failure happened, they usually spoke out, saying, for example, “it didn't work,” “something is wrong,” “what?!”, “not again.” I labelled the most common errors as syntax errors, logic errors, arithmetic errors, and variable errors. Syntax errors occurred when the code was not valid according to the rules of the programming language (e.g., using a wrong coding block). Logic errors occurred when the code was written correctly from a syntax perspective but did not Figure 4.3.9 Sample of Input Error 102 produce the desired results because the logic of the program was flawed (e.g., an unnecessary infinite loop). Numerical errors arose when students conducted tests using various numerical values for blocks, such as exceeding the maximum allowable numeric value. For instance, in the case of the power input, it was designed to accept values within the range of 0 to 10. However, when students attempted to input a value such as 23, the system would automatically revert it to the maximum allowed value of 10. While the code could technically be executed, it wouldn't produce the desired output or behavior due to the input value exceeding the specified range. Errors associated with incorrect input occurred when students used incorrect types of input in their programs, potentially leading to unforeseen outcomes. For instance, as depicted in Figure 4.3.9, in the Walker project, Carl attempted to link a microphone to the loop, possibly intending for the loop to halt in response to a sound input. However, it should be noted that the loop could only be defined by a numerical input. When Carl executed this code, the robot experienced a complete standstill, exhibiting no signs of movement or activity. In these robotic activities, students demonstrated a level of independence in recognizing errors and challenges and taking corrective measures without external guidance or intervention. They regularly found themselves confronted with an array of errors. These errors spanned from relatively straightforward design glitches to intricate coding mistakes. Some of these errors were immediately identifiable by students with a conspicuous performance of their robots. However, there existed errors in coding that proved more elusive, requiring a deeper level of investigation and diagnostic effort. 103 4.4.2 Students’ Responses to Errors The study reveals how students responded to errors or setbacks with a positive attitude. Across cases, students frequently made comments such as “get is work,” “let’s get it fixed,” or, “I'll fix it,” when there was a problem, indicating their attitude of persistence by stating their intention to refine and improve their work. Sylvia encountered a challenge related to the functionality of a robotic hammer she was working on. The conversation in the interview captures her problem-solving process and reveals key insights into her responses to errors. Sylvia: It's not working. Oh… no… I figured out why. It's not this part [she checked the motor and gears] Me: What did you figure out? Sylvia: um I figured out why it wasn't working…one of the pieces that like pushes down…so it fell off. Me: How did you figure it out? Sylvia: I looked inside… I'm like ‘oh my gosh, it fell off’. Me: When you figure that out… how did you feel about that? Sylvia: I'm…kind of like … ‘oh my God’. It takes a while. The conversation reveals that Sylvia not only identified the problem but also described how she felt and what actions she took to address it. When asked about her feelings after figuring out the issue, Sylvia expressed a sense of accomplishment and relief: “I'm like really … kind of like finally figured it out … oh my God…it takes a while.” This statement underscores the time and 104 effort invested in the problem-solving process, emphasizing persistence and determination in the face of challenges. During a video recall interview, I asked Carl about why he laughed when the car they were building broke. Carl attributed his laughter to the seemingly explosive nature of the incident, humorously mimicking an exploding gesture and saying “boom.” However, as our conversation delved deeper into whether other types of mistakes such as in practicing piano would evoke a similar response, Carl concluded by stating, “Only if it's not important.” The laughter in response to a broken robot seemed to stem from the perception that these mistakes were not overly critical in the after-school robotics activities, especially when compared to other learning activities labeled as more “important”. In the absence of explicit communication on “serious learning” in the activity, Carl' main motivation appeared to be the enjoyment derived from problem-solving and experimentation. Throughout the study, students did not appear to negatively judge each other or themselves when anyone made mistakes, or did not solve the challenge at the end of class. They exhibited a commendable tolerance for mistakes and failures and remarkable persistence in handling complex situations, even when it entailed the possibility of making additional errors. This persistence and commitment to overcoming setbacks was consistently observed among all subcases. For example, in the Clock project, Me: What is going on? Can you tell me? Jason: I don’t know. It is going [on] forever. 105 Me: What happened then? Jason: Then [it] broke. Me: What did you decide to do? Jason: Fix it again. Me: Again? Is this not the first time you have fixed it? Jason: No. Princeton: [he interrupted Jason] Not the first time. We fixed it already a lot of times. Despite multiple attempts without clear signs of progress, Jason insisted on fixing the identified error leading to non-stop spinning. The fact that it was not the first time that they fixed the robot showed their collaborative effort in dealing with failures and their willingness to persevere in the face of challenges. In this context, students not only demonstrated high tolerance towards mistakes and failures but also showed persistence in handling complex situations, even if it meant that they might make additional mistakes. This insight suggests that students embraced a relatively lighthearted attitude towards failures and errors within this learning environment, recognizing that the after-school robotics program offered a space for experimentation. It is crucial to acknowledge that not all student pairs displayed identical attitudes towards setbacks. Examining the specific case of Carl and Kent revealed variations in their responses. When questioned about their thoughts on the fixes implemented, Kent subtly hinted at feelings of frustration or disappointment regarding the corrective measures. On the other hand, Carl openly acknowledged making coding mistakes, stating that “[I am] not correct all the time.” This candid admission suggests that Carl viewed errors as an inherent part of the activity and deemed them 106 acceptable. Those who navigated challenges effectively did so by actively engaging in constructive communication and diplomatically resolving conflicts within their teams. This collaboration not only fostered a positive team dynamic but also contributed to the problem-solving process as they tried to work out a solution. In contrast, instances where frustration or disappointment were hinted at, as seen in Kent's case, underscored the importance of addressing emotional responses to setbacks as a critical aspect of problem solving learning. While high tolerance for mistakes and persistent problem-solving were consistently observed among subcases, individual attitudes towards setbacks varied. The recognition and acceptance of errors as integral to the problem-solving process played a pivotal role in shaping the students' overall success in navigating the complexities of robotics problem-solving. However, not all pairs showed the same attitudes towards setbacks. For Subcase 4 (Carl and Kent), when asked about their thoughts on these fixes, Kent hinted at frustration or disappointment about fixes. Carl acknowledged making coding mistakes— “not all the time [I am] correct”, hinting that these errors were acceptable. Most subcases held the perspective that redoing was an indispensable component of achieving success. An illustrative instance of her positive mindset can be seen in Syliva’s Clock project experience. When confronted with building errors that she described as “tricky,” she took a proactive approach by dismantling the robot, without hesitation. Her willingness to deconstruct her creation in the pursuit of solutions exemplified her persistence and commitment to achieving 107 her goals. Princeton, when asked about his perspective on the mistakes he made, responded with enthusiasm, stating, “It is fun because I can fix it.” I interpreted that he found enjoyment in the process of troubleshooting and viewed errors as intriguing challenges to overcome. For him, working with robotics might not be just about achieving a flawless result; it was about the excitement of identifying and resolving issues. His outlook underscored the idea that mistakes were valuable opportunities for refinement and improvement. Tania, in her Walker project, faced challenges that she humorously summarized by saying, “Well, I'm just out of time.” Her lighthearted response demonstrated her ability to maintain an optimistic attitude despite not achieving complete success in the project. 4.5 Chapter Summary A key finding of this study is the iterative nature of how these students approached problem-solving during the Challenge stage of the activities. This iterative process, characterized by continuous cycles of idea generation, experimentation, and evaluation, played a central role in their problem-solving strategies. Their paths diverged as they engaged in various actions throughout the activity to build and code robots for specific goals. Despite these differences, all students engaged in an iterative process, conceiving an initial idea, translating it into experiments, and then evaluating the results. It is worth noting that most students revisited the original problem (i.e., understanding phase) from various angles, leading to the emergence of new ideas and prompting renewed iterative cycles throughout the problem-solving process. During these iterations, students employed a wide range of problem-solving strategies, including decomposing, trial-and-error, and self-assessment. Decomposing involved breaking down 108 complex problems into more manageable, smaller tasks, making them easier to tackle. Trial-and-error was evident as students experimented with various configurations or sequences to discover effective solutions. Additionally, self-assessment allowed students to reflect on their progress, identify areas for improvement, and refine their approaches. These problem-solving techniques were prevalent throughout the study and contributed to the students' problem-solving in robotics activities. Further, in robotics activities, students employed various forms of representation of mathematical concepts such as symbolic representations, drawings, constructed objects, coding, and oral/mental activities. These multidimensional explorations allowed students to translate abstract mathematical concepts into tangible representations. It is noteworthy, while evidence showed that they used a wide range of mathematical skills and knowledge to solve problems, students did not fully recognize the extent to which their actions involved mathematics. In these non-routine activities, students actively took charge of their problem-solving, such as setting goals, carrying out actions, and evaluating their work. Their enthusiasm for establishing goals that resonated with their personal aspirations showcased the opportunity for decision-making that these activities provided. Moreover, the students exhibited persistence in the face of setbacks and errors. The evidence demonstrates that students used errors and failures as building blocks to refine their strategies and enhance their problem-solving solutions. In the next chapter, I discuss my findings in relation to the theoretical framework outlined in Chapter Two and in light of current literature and research on the use of robotics in mathematics 109 education. I also outline implications for future research and practice in mathematics education. 110 Chapter 5: Discussion In this chapter, I discuss my findings related to two research questions: 1) How did young children use mathematics as they engaged in robotics activities? and 2) What self-regulated learning opportunities did robotics afford as young children engaged in problem-solving activities? I begin by scrutinizing these findings in relation to the existing literature, followed by an exploration of their practical implications for after-school settings and math classrooms. Lastly, I will engage in a discussion regarding recommendations for further research in this area. 5.1 Children’ Use of Mathematics in After-school Robotics My study illuminated a consistent iterative problem-solving approach shared by four subcases in a total of 24 non-routine robotics sessions, with 11 distinct activities. The students' engagement in these activities was marked by continuous cycles of four stages: understanding the problem, ideation, experimentation, and assessment. Throughout this iterative process, students did not limit themselves to a fixed idea or design. Instead, they embraced a more exploratory approach, continuously refining and testing multiple iterations of their ideas at each stage of development. This finding contributes to the problem-solving model introduced by Polya (1971, 1981), which has significantly influenced mathematics education over the years. By demonstrating the iterative nature of problem-solving and the importance of exploring multiple iterations of ideas, my study reinforces the idea that problem-solving is not a linear process but rather a dynamic and iterative one, where students continuously refine their approaches and strategies until they arrive at a satisfactory solution. In my study, students were actively involved in a continuous process of problem comprehension and redefinition. As they iterated through various stages of 111 development, students demonstrated a growing understanding of the problem they were tasked with, including its nuances, complexities, and limitations. The findings of my study highlight aspects of students’ problem-solving that mirror computational thinking as articulated in Wing (2008) and Carnegie Mellon (2023). According to these authors, the idea of computational thinking is to approach problem-solving in a systematic and logical way, similar to how a computer scientist might tackle challenges. This encompasses the decomposition of intricate problems into manageable components, the abstraction of situations, pattern recognition, and the application of logical and algorithmic reasoning. In my investigation, students approached robotics challenges by breaking them down into more manageable and comprehensible components. This strategy supported students in my study to address each aspect of the problem systematically, focusing on one piece at a time. By adopting this methodical approach, students engaged in a process that mirrors computational thinking strategies, enabling them to analyze complex problems systematically and develop effective solutions. Additionally, students demonstrated algorithmic thinking through the common use of trial-and-error in both constructing and coding stages. These observed strategies not only underscore the pragmatic aspects of computational thinking but also emphasize how students, within the context of robotics, actively engage with its foundational principles. In this study, students demonstrated remarkable persistence when faced with non-routine challenges. They derived joy and seized opportunities from their errors. This finding suggests that, within this particular context, students were not solely fixated on finding the "right" answer. Instead, they embraced a mindset that prioritized exploratory experiences, seeking out unique 112 and meaningly solutions for themselves. This mindset to problem-solving aligns closely with the concept of “problem-solving habits of mind” as coined by Rigelman (2007). The author argues that in order to foster problem-solving habits of mind, students require exposure to working with various situations that they “problematize with the goal of understanding and developing solution methods that make sense to them” (Hiebert et. al., 1996, p.19, as cited in Rigelman, 2007, p.308). In my study, the non-routine activities provided students with experiences that encouraged them to actively engage with problems, with the aim of understanding and devising solution methods that resonated with their own reasoning processes. Through iterative and exploratory endeavors, students were able to develop a deeper understanding of problems and to devise solutions that were meaningful and creative. My study revealed a noticeable synergy between robotics and mathematics. Robotics in these activities not only provided students with practical applications of mathematical concepts but also stimulated their curiosity and creativity, albeit implicitly. The integration appeared to create a learning environment that encouraged engagement in the meaning-making process. According to constructionists such as Schoenfeld (2019) and Papert (1980), it is essential to provide students with meaningful learning experiences that enable them to apply mathematical concepts in authentic problem-solving scenarios. Further, the interactive learning experiences, as observed in this study, empowered students to express their understanding of mathematical concepts through multiple representations. The importance of using multiple representations in mathematics learning is highlighted by the analogy of blindfolded individuals interacting with an elephant (Tripathi, 2008). According to the author, just as each unique perspective provided limited but valid insights into the elephant's nature, the varied representations of mathematical 113 concepts offered in these robotics activities could provide opportunities for students to gain, or at the very least, demonstrate a diverse understanding of embedded mathematics. Throughout the phases of designing, programming, and troubleshooting the robot, students’ engagement with mathematics included computation and more, reflecting a broader range of mathematical thinking. They utilized numerical operations such as addition, subtraction, and multiplication to calculate time, speeds, and quantities essential for the robot's functionality. Moreover, they applied geometric concepts such as angles to design the movements of the robot. Furthermore, students engaged in pattern recognition and analysis to identify irregularities from observed behaviors of the robot. However, students in my study perceived their use of mathematics narrowly, primarily attributing it to basic arithmetic. The disconnection could potentially be attributed to the fact that the program was not explicitly designed with a central focus on mathematics. The instructors emphasized that the primary objective of the robotics curriculum was to cultivate students' proficiency in constructing fundamental engineering structures and acquainting them with essential coding principles. Furthermore, Schoenfeld (2018) argued that students may have a limited understanding of the scope of mathematics. Students may have limited knowledge of the various mathematical concepts and skills involved in cooking, video gaming, and many daily events that are beyond traditional classroom settings. This may partially explain why the children in my study did not view their experiences as mathematical. Nevertheless, the significant gap between students' perception of their use of mathematics and the actual mathematical applications they engaged in suggests untapped opportunities to explore 114 the connections among robotics, computer programming, and mathematics. According to Papert (1980), one potential benefit of helping students link their learning in math to authentic activities is the enhancement of their overall mathematical understanding and engagement. The connection between math and robotics would allow them to witness firsthand how mathematical ideas are not abstract but rather practical tools that can be used to solve real problems. When students see the real-world applications of mathematical concepts, they could be more likely to become motivated and interested in the subject. 5.2 Children’s Self-Regulated Learning in Robotics The findings of my research have illuminated that students as young as six years old can self-regulate their learning within the context of robotics. These students, in their problem-solving endeavors, actively engaged with the learning space including the material, the instruction of the activity, the other students and the instructor. The rich interaction between peers observed in the two cases with student pairs within the learning environment underscores that self-regulation in robotics activities was not a solitary endeavor and it did not occur in isolation of its surrounding context. Rather, my findings highlight the intricate ways in which individuals exercise agency, as they interpret and respond to the contextual factors at play, both independently and collaboratively. The findings also resonate with the principles of situated self-regulated learning proposed by Zimmerman (2000) and (Butler et al., 2017), which emphasizes the influence of environmental factors on individuals' learning strategies. In the context of robotics activities, the findings underscore how students' interactions with their learning context played a role in shaping their 115 problem-solving approaches, goal-setting strategies, and self-monitoring behaviors. As an illustration, students were observed collaborating with peers to brainstorm ideas; the open-ended instructions markedly influenced students' approaches and strategies to the tasks; additionally, the feedback provided by both the robot and peers proved pivotal in molding students' assessment process. In essence, the tasks, instructions, and feedback collectively influenced students' engagement in autonomy, collaboration, and persistence in problem-solving within the robotics context. In my study, students demonstrated agency by actively setting personalized goals for themselves. Prior to initiating a robotics project, students engaged in a process of interpreting the tasks outlined in the activity and establishing a purpose for their robot. This process involved consideration of the objectives of the project, as well as the specific functionalities and capabilities they wished their robot to possess. As students engaged with the tasks and challenges presented, they demonstrated a dynamic process of goal-setting, monitoring, and adaptation—a hallmark of effective self-regulation. Successful participation in these robotics activities frequently depended on students' capacity to select, to adapt, or even to create personalized goals tailored to the problem at hand. These findings resonate with the work of Black and William (2009), who advocate that students become more adept at self-regulating their performance successfully when they have a clear understanding of what they are striving to accomplish. Consequently, these robotics activities provided both opportunities and immediate feedback for students in terms of regulating their own learning. According to Perry (2006), encouraging students to make meaningful choices and decisions towards a learning goal fosters autonomy. This suggests that in robotics activities, where students had the freedom to choose which 116 problems to tackle, make decisions about materials, and determine how to assess and present their solutions, students were empowered to exert deliberate control over the challenges they encountered. This autonomy may foster a sense of ownership and responsibility in students, allowing them to take charge of their learning experiences—an important quality applicable to various aspects of their learning, including in mathematics. My study revealed that, in the open-ended learning environment, students actively engaged in the assessment process throughout problem-solving. Instructors made it clear to students that these activities had no definitive right or wrong answers, and there were no exams attached to them. The absence of formal evaluation and open-ended design of the activity seemed to foster an environment that encouraged students to participate in self-assessment. As students programmed and executed their code, the robots provided immediate responses, offering insights into the success or challenges of their current solution. Additionally, in paired settings, students exchanged feedback, contributing to a collaborative learning environment. This feedback loop, involving both robotic responses and human interactions with instructors and peers, played an active role in supporting students' self-assessment. It seemed to help students evaluate their progress, identify areas for improvement, and adapt their strategies accordingly. This metacognitive process aligns closely with SRL principles, as students became more aware of their strengths and weaknesses and develop strategies to optimize their learning experience. This finding aligns with the insights from the work of Brookhart (2008), who argues that through self-assessment, students not only evaluate their own performance but also learn how to generate valuable information for themselves. The author claims that this self-generated information serves as a guide for their further activity, reinforcing the idea that self-assessment is a dynamic 117 process that empowers students to take charge of their own learning. This study underscored the dynamic nature of shared regulation in robotics. Students worked in pairs, sought and provided constructive feedback to each other, supporting each other to engage in strategic actions centered around problem-solving. This collaborative dimension added depth to the SRL process, where students jointly navigated challenges, celebrated successes, and iteratively refined their strategies. In this setting, students not only assessed their individual contributions but actively engaged in shared reflection on their collective progress. The finding aligns with the assertion of Hadwin et al. (2018) that diverse learners contribute various perspectives, co-constructing a more comprehensive sense of purpose. Their joint reflection on challenges, coupled with expressions of disappointment or satisfaction, illuminated a socially shared regulation process put forth by Zimmerman (2000) and Järvelä and Hadwin (2013). 5.3 Implications for Practice This study offers insight into how students leveraged mathematics and self-regulated learning in an after-school robotics context, which was characterized by its unique curriculum and teaching approach. The program emphasized exploration and discovery, with minimal formal evaluation components. The participants, students with a genuine interest in robotics, worked on diverse projects with instructors who served as facilitators. Understanding this contextual backdrop is integral to the nuances of the findings reported herein. In this section, I aim to explore implications and recommendations to enhance mathematical problem-solving and Self-Regulated Learning (SRL) in after-school programs. Additionally, I will discuss the potential implications of integrating robotics into math classrooms. Future research should continue to investigate this 118 learning approach to provide further insights into its impact on students' academic and personal development within the broader educational landscape. 5.3.1 Incorporating Non-Routine Tasks in Robotics Activities My study suggests that the introduction of open-ended and non-routine robotics tasks could enhance the iterative process of problem-solving. These activities could offer young students an engaging space to embrace their flexibility and creativity. Rather than following prescribed steps, students would be immersed in exploring alternative approaches, experimenting with different strategies, and iterating upon their solutions based on feedback and reflection. Additionally, by removing the constraints of predetermined outcomes, open-ended activities could encourage students to think critically to devise unique solutions to complex problems. Furthermore, non-routine robotics tasks have the potential to establish a framework for normalizing errors and challenges in problem-solving. Given the nature of these tasks, which frequently introduce students to novel and unfamiliar situations, there is an increased likelihood of encountering errors along the way. In this way, non-routine robotics tasks could serve as an ideal platform that encourages students to engage in experimentation, trial and error, and creative problem-solving. By engaging in non-routine robotics tasks, students would be presented with opportunities to learn from errors and to enhance the persistence needed to navigate through the unpredictable situations presented by these tasks. 5.3.2 Embracing Chaos in Robotics Problem-Solving In the context of robotics problem-solving, while task understanding is undoubtedly critical, 119 expecting students to have a full and complete understanding at the outset might not be realistic. Instead of expecting immediate mastery, instructors should allow students to explore and learn through iterations. My study suggests that young students would frequently encounter challenges rooted in a lack of comprehensive or even partial understanding of the given task. For instance, in the clock project, students’ initial solutions included allowing the clock to move really fast and allowing it to move counter-clockwise. Some solutions devised by the students deviated significantly from what might be considered “correct” in the conventional sense (i.e., inaccurate, and wrong direction). However, over time these students created a unique understanding that often emerged and evolved through the act of problem-solving and iteration itself. With robotics, even though students' initial endeavors may lack perfection, if instructors allow the focus of the learning to shift from pre-set solutions towards students’ solutions, it could encourage students to learn from mistakes, make necessary adjustments, and continuously refine their work to achieve improved outcomes. Placing emphasis on incremental progress, rather than fixating on a singular correct or optimal solution, could hold significant value in engaging students in the iterative problem-solving process itself. Starting with a sense of purpose is indeed key for young learners in problem-solving (Polya,1981), and a messy start could serve a role to reinforce the sense. Enabling students to immerse themselves in ambiguous situations, even at the risk of experiencing multiple errors and failures, is advisable in the context of robotics to foster such sense. Open-ended activities invited students to dive into exploration with curiosity and enthusiasm, allowing them to actively immerse themselves in the process of problem-solving. As students navigate through various 120 possibilities, students could gain clarity on their goals, guiding their efforts towards achieving meaningful outcomes, and ultimately a more defined sense of purpose. 5.3.3 Supporting SRL in Robotics in After-School Programs The integration of robotics into after-school programs could serve as a catalyst for promoting effective SRL among young students, particularly in goal-setting and self-assessing. Enabling students to set goals to align with their personal aspirations underscored the decision-making opportunities inherent in these activities. A key strategy to support goal setting could involve enabling students to identify their own goals. By allowing them to set objectives that are personally relevant, motivation could be enhanced, contributing to a more meaningful learning experience. Moreover, embracing the concept of breaking down overarching goals into smaller, manageable tasks could reinforce students' ability to navigate the pathway toward achieving their ultimate objectives. This incremental approach could cultivate a sense of accomplishment with each milestone achieved, therefore equipping young learners to persist through inevitable mistakes and setbacks encountered along the way. Students should be encouraged to reflect on their progress and areas for improvement. Self-assessment holds potential in empowering students to make informed and strategic decisions throughout their engagement with robotics projects. In the robotics activities included in this study, students gained access to continuous visual feedback, providing an understandable means for assessing their progress. This feedback loop could guide students to monitor their advancements and identify areas for improvement. An essential aspect of facilitating effective self-assessment could involve allowing students the flexibility to customize their own evaluation 121 criteria. Each student could bring unique perspectives, experiences, and priorities to the table, and allowing them to shape their own criteria would acknowledge and respect their individual needs. When students are involved in determining the criteria for measuring their own progress, they could gain insight into their strengths and areas for improvement and become invested in their own learning process. Advocating for students to work in pairs holds another promise to enhance SRL. The collaborative arrangement could foster an environment where mutual support is readily exchanged, thereby contributing to the continual development and strengthening of students’ SRL capabilities. When working in pairs, students could have the opportunity to engage in collaborative problem-solving, where they could share ideas, perspectives, feedback, and strategies. By actively participating in discussions and decision-making processes, students could become more aware and reflective of their learning goals and progress. This metacognitive awareness is essential for developing effective self-regulation skills. 5.3.4 Connecting Robotics to Mathematics in School Classrooms Although my study took place in an after-school setting, my study yields insights that are relevant to the integration of robotics into mathematics classrooms. By incorporating robotics, students could apply mathematical principles in a tangible context. If students have opportunities to visually perceive that they are learning mathematics through the activities, it could further enhance the educational impact. This integration could make mathematics relevant and interesting to students, and could equip them with valuable problem-solving skills reflective of computational thinking. 122 My research findings revealed that students often viewed mathematics narrowly, primarily associating it with arithmetic. To address this limitation, my study suggests that incorporating robotics into the mathematics curriculum could encourage interdisciplinary thinking, as students could see first-hand the connections between mathematics, engineering, computer programming, and real-world problem-solving. Through engaging with robotics, students could be empowered to explore relationships between mathematics and other domains. Thus, the incorporation of robotics into the mathematics curriculum could serve as a catalyst for cultivating a holistic appreciation for the interdisciplinary nature of mathematical knowledge. Furthermore, enabling students to use diverse representations in their problem-solving endeavors could enhance their engagement in in mathematical problem-solving . It is advisable to nurture an environment that appreciates the value of each individual representation which is accessible to the students, including symbolic expression, physical models and coding scripts. These representations highlight their interconnectedness with various fields. These instructional efforts could further instill the notion that mathematical problem-solving is a multifaceted process, which goes beyond computation and fixed answers. 5.4 Implications for Future Research While my study provides insights into how students used mathematics and self-regulated learning in an after-school robotics context, its inherent limitations, as well as the findings, point towards the need for additional research. I am inviting further exploration within the field of robotics in mathematics education rather than recommending direct application within an educational context. 123 Conducting comparative studies across various robotics programs, both within after-school and in-school settings, could offer a more nuanced perspective. By examining how students engage with mathematics and self-regulated learning in different instructional environments, researchers might uncover patterns and variations contributing to a more comprehensive framework. One must consider how the findings might differ if students from the same program were informed from the outset that the activities were specifically intended as math exercises and would be scored at the end of the program. Introducing a control group with this altered awareness level could potentially reveal whether the students' engagement and self-regulated learning behaviors are influenced by such knowledge. In addition, longitudinal studies could shed light on the sustainability and long-term impact of integrating robotics in mathematics and self-regulated learning. By tracking students' progress over an extended period, researchers might gain insights into the retention of skills and the transfer of learning to other contexts. Such studies would offer an understanding of the lasting benefits of robotics education in fostering mathematical understanding and self-regulated learning skills. My study focused on lower grades participants’ experiences with robotics activities. Future studies that explore factors such as age, prior knowledge, and learning preferences could provide valuable insights into how individual differences influence students' engagement with these concepts. Such studies could lead to the development of personalized approaches that cater to diverse student needs and enhance their learning outcomes. 124 Future research should aim to include a wider array of robotics programs and explore their effectiveness in various in-school settings. Additionally, longitudinal studies would provide a more comprehensive understanding of how students of different developmental stages use mathematics and self-regulated learning in problem-solving. By focusing on such issues, researchers can gain deeper insights into the role of mathematics and self-regulated learning in problem-solving, leading to a more comprehensive understanding of effective educational practices. 5.5 Conclusion By examining how young children used mathematics and engaged in self-regulated learning during robotics activities, my study contributes to the growing body of knowledge on educational robotics and mathematics education. Regarding my first research question about how young children used mathematics as they engaged in robotics activities, my findings illustrate that the process of problem-solving in these activities was not a linear, locked step-by-step progression; rather, it constitutes a dynamic and iterative journey. Students actively participated in an ongoing interplay among various stages, with the flexibility to move freely back and forth between them as they navigate their problem-solving endeavors. My findings reveal that the young children in this study applied various mathematical concepts and skills, and to solve problems. Students communicated their mathematical knowledge and solutions through multiple representations, including iconic representations peculiar to coding scripts, paper and pencil symbols and drawings, verbalization of mental calculation, and so forth. 125 In solving problems, students broke down complex problems into smaller, more manageable parts, identified patterns and relationships, developed step-by-step coding commands, and used trial-and-error to troubleshoot problems, all of which reflected principles of computational thinking. Even though robotics activities provided an engaging context for children to apply their mathematical knowledge and computational thinking skills, they did not fully recognize they were doing so. To address my second question regarding self-regulated learning (SRL) in action, my study findings revealed that robotics activities consistently presented opportunities for young students to practice various aspects of self-regulated learning. The dynamic nature of robotics allowed children to readily determine whether their solutions 'worked or not,' diminishing the need for consulting an outside authority. With non-routine tasks at hand, students were empowered to make choices and evaluate progress continuously. This environment provided students with opportunities to assume ownership of their learning within these activities. Within this framework, students engaged in vital components of self-regulated learning (SRL), including setting goals, self-assessment, and persevering through challenges encountered during problem-solving tasks. Moving forward, it is vital to continue exploring the synergies between SRL, mathematics, and robotics across diverse educational settings. By deepening our understanding of the connections between these domains, and hence providing learning experiences that highlight the meaning-making process, mathematics educators could empower students by nurturing self-regulation, enhancing problem solving skills, and preparing them for an ever-evolving world. 126 References Ackermann, E. (2001). Piaget’s constructivism, Papert’s constructionism: What’s the difference? Future of Learning Group Publication, 5(3),438. Retrieved from http://learning.media.mit.edu/content/publications/EA.Piaget%20_%20Papert.pdf Alesandrini, K., & Larson, L. (2002). Teachers bridge to constructivism. The Clearing House, 75(3), 118-121. https://doi.org/10.1080/00098650209599249 Ardito, G., Mosley, P., & Scollins, L. (2014). WE, ROBOT: Using robotics to promote collaborative and mathematics learning in a middle school classroom. Middle Grades Research Journal, 9(3), 73. Atman, C. J., Adams, R. S., Cardella, M. E., Turns, J., Mosborg, S., & Saleem, J. (2007). Engineering design processes: A comparison of students and expert practitioners. Journal of Engineering Education, 96(4), 359–379. Atmatzidou, S., & Demetriadis, S. (2014). How to support students’ computational thinking skills in educational robotics activities. Paper presented at the Proceedings of 4th International Workshop Teaching Robotics, Teaching with Robotics & 5th International Conference Robotics in Education. Atmatzidou, S., & Demetriadis, S. (2016). Advancing students’ computational thinking skills through educational robotics: A study on age and gender relevant differences. Robotics and Autonomous Systems,75, 661–670. Baek Y., Wang S., Yang D., Ching Y.-H., Swanson S., Chittoori B. (2019). Revisiting second graders’ robotics with an Understand/Use-Modify-Create (U2MC) strategy. European Journal of STEM Education, 4, 1–12. https://doi.org/10.20897/ejsteme/5772 127 Bandura, A. (2000). Exercise of human agency through collective efficacy. Current Directions in Psychological Science, 9(3), 75–78. Barak, M., & Zadok, Y. (2009). Robotics projects and learning concepts in science, technology and problem solving. International Journal of Technology and Design Education, 19(3), 289-307. doi:10.1007/s10798-007-9043-3r5 BC Ministry of Education (2016). Curriculum overview. In BC’s new curriculum. Retrieved from https://curriculum.gov.bc.ca/curriculum/overview Benitti, F. B. V. (2012). Exploring the educational potential of robotics in schools: A systematic review. Computers & Education, 58(3), 978–988. Bharatharaj, J., Pepperberg, I. M., Sasthan Kutty, S. K., Munisamy, A., & Krägeloh, C. (2023). Exploring the utility of robots as distractors during a delay-of-gratification task in preschool children. Frontiers in Robotics and AI, 10, 1001119-1001119. https://doi.org/10.3389/frobt.2023.1001119 Blanchard, S., Freiman, V., & Lirrete-Pitre, N. (2010). Strategies used by elementary school children solving robotics-based complex tasks: Innovative potential of technology. Procedia, Social and Behavioral Sciences, 2(2), 2851-2857. https://doi.org/10.1016/j.sbspro.2010.03.427 Boaler, J. (1993). The role of contexts in the mathematics classroom: Do they make mathematics more “real”? For the learning of mathematics, 13(2), 12-17. Boaler, J. (2013). Ability and mathematics: The mindset revolution that is reshaping education. Forum for Promoting Comprehensive Education, 55(1), 143. 128 Boaler, J. (2022). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages, and innovative teaching. 2nd Edition. San Francisco, CA: John Wiley & Sons. Boaler, J., Dieckmann, J. A., LaMar, T., Leshin, M., Selbach-Allen, M., & Pérez-Núñez, G. (2021). The transformative impact of a mathematical mindset experience taught at scale. Frontiers in Education (Lausanne), 6. https://doi.org/10.3389/feduc.2021.784393 Boekaerts, M. (1991). Subjective competence, appraisals and self-assessment. Learning and Instruction, 1(1), 1-17. doi: 10.1016/0959-4752(91)90016-2 Bransford, J. (2000). How people learn: Brain, mind, experience, and school (Expand ed.). Washington, D.C: National Academy Press. Brender, J., El-Hamamsy, L., Bruno, B., Chessel-Lazzarotto, F., Zufferey, J. D., & Mondada, F. (2021). Investigating the role of educational robotics in formal mathematics education: The case of geometry for 15-year-old students. Technology-enhanced learning for a free, safe, and sustainable world (pp. 67-81). Springer International Publishing. https://doi.org/10.1007/978-3-030-86436-1_6 Burris. A. C. (2005). Understanding the math you teach: Content and methods for prekindergarten through Grade 4. Prentice Hall: Pearson. Butler, D. L., Perry, N. E., & Schnellert, L. (2017). Developing self-regulating learners. Toronto: Pearson. Butler, D. L., & Cartier, S. C. (2018). Case studies as a methodological framework for studying and assessing self-regulated learning. In D. Schunk & J. Greene (Eds.), Handbook of self-regulation of learning and performance (2nd ed.), pp. 352-369. NY: Routledge. 129 Butler, D. L., & Schnellert, L. (2020). Negotiating Meaning and Engagement: Socially Shared Strategic Processing. In Dinsmore, D. L., Fryer, L. K., & Parkinson, M. M., Handbook of strategies and strategic processing: Conceptualization, intervention, measurement, and analysis (pp. 63-81). NY: Routledge. Carnegie Mellon.(2023). Center for Computational Thinking. Retrieved from http://www.cs.cmu.edu/~CompThink/ Cassidy, S. (2015). Resilience building in students: The role of academic self-efficacy. Frontiers in Psychology, 6, 1781-1781. https://doi.org/10.3389/fpsyg.2015.01781 Chen, G., Shen, J., Barth-Cohen, L., Jiang, S., Huang, X., & Eltoukhy, M. (2017). Assessing elementary students’ computational thinking in everyday reasoning and robotics programming. Computers & Education,109, 162–175. Cherniak, S., Lee, K., Cho, E., & Jung, S. E. (2019). Child-identified problems and their robotic solutions. Journal of Early Childhood Research: ECR, 17(4), 347-360. https://doi.org/10.1177/1476718X19860557 Chi, M. T. H., & Wylie, R. (2014). The ICAP framework: Linking cognitive engagement to active learning outcomes. Educational Psychologist, 49(4), 219-243. https://doi.org/10.1080/00461520.2014.965823 Chin, D. B., Blair, K. P., Wolf, R. C., Conlin, L. D., Cutumisu, M., Pfaffman, J., & Schwartz, D. L. (2019). Educating and measuring choice: A test of the transfer of design thinking in problem solving and learning. Journal of the Learning Sciences, 28(3), 337–380. https://doi.org/10.1080/10508406.2019.1570933 Christensen, R., Knezek, G., & Tyler-Wood, T. (2015). Alignment of hands-on STEM engagement activities with positive STEM dispositions in secondary school students. 130 Journal of Science Education and Technology, 24(6), 898-909. doi:10.1007/s10956-015-9572-6 Chuang, C., & Cheng, H. H. (2022). Learning algebra with robotics in math classes. Paper presented at the 2022 18th IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications (MESA), Taipei, Taiwan, 2022, pp. 1-6, doi: 10.1109/MESA55290.2022.10004399. Cobb, P., Wood, T., Yackel, E. (1991). A Constructivist Approach to Second Grade Mathematics. In: Von Glasersfeld, E. (eds) Radical Constructivism in Mathematics Education. Mathematics Education Library, vol 7. Springer, Dordrecht. https://doi.org/10.1007/0-306-47201-5_8 Cotič, M., & Zuljan, M. (2009). Problem-based instruction in mathematics and its impact on the cognitive results of the students and on affective-motivational aspects. Educational Studies, 35(3), 297-310. doi:10.1080/03055690802648085 Creswell, J. W. (1998). Qualitative inquiry and research design: Choosing among five traditions. Sage Publications, Inc. Cuban, L. (1986). Teachers and machines: The classroom use of technology since 1920. New York: Teachers College Press. Cuban, L. (2001). Oversold and underused: Computers in the classroom. Cambridge, Mass: Harvard University Press. Demetroulis, E. A., Theodoropoulos, A., Wallace, M., Poulopoulos, V., & Antoniou, A. (2023). Collaboration skills in educational robotics: A methodological Approach—Results from two case studies in primary schools. Education Sciences, 13(5), 468. https://doi.org/10.3390/educsci13050468 131 Docan‐Morgan, T. (2015). The participation log: Assessing students' classroom participation. Assessment Update, 27(2), 6-7. doi:10.1002/au.30016 Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., Pagani, L. S., Feinstein, L., Engel, M., Brooks-Gunn, J., Sexton, H., Duckworth, K., & Japel, C. (2007). School readiness and later achievement. Developmental Psychology, 43(6), 1428–1446. https://doi.org/10.1037/0012-1649.43.6.1428 Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1/2), 103-131. https://doi.org/10.1007/s10649-006-0400-z Dweck, C. (2008). Mindset: The New Psychology of Success. New York: Ballantine Books English, L. D., & Halford, G. S. (2009). Mathematics education: Models and processes. New York: Routledge English, L. D. (2016). STEM education K-12: Perspectives on integration. International Journal of STEM Education, 3(3), 1. https://doi.org/10.1186/s40594-016-0036-1 English, L. D., (2022). Multidisciplinary Modelling in a Sixth-Grade Tsunami Investigation. International Journal of Science and Mathematics Education, 21(Suppl 1), 41-65. https://doi.org/10.1007/s10763-022-10303-4 English, M. C., & Kitsantas, A. (2013). Supporting student self-regulated learning in problem- and project-based learning. Interdisciplinary Journal of Problem-Based Learning, 7(2). Retrieved from: http://dx.doi.org/10.7771/1541-5015.1339 Eraikhuemen, L., & Oteze, I. K. (2015). An empirical study of the perception of undergraduates of Nigerian females' participation in science, technology, and mathematics. Journal of International Women's Studies, 16(3), 33. 132 Fissore, C., Marchisio, M., Roman, F., & Sacchet, M.Development of problem solving skills with maple in higher education. Maple in mathematics education and research (pp. 219-233). Springer International Publishing. https://doi.org/10.1007/978-3-030-81698-8_15 Foss, A., Wilcoxen, C., & Rasmus, J. (2019). The academic and behavioral implications of robotics in the classroom: An elementary case study. Technology and Innovation, 20(3), 321-332. https://doi.org/10.21300/20.3.2019.321 Gagliardi, M., Bartolucci, V., Scaradozzi, D. (2021). Educational robotics at primary school with Nintendo Labo. In: Scaradozzi, D., Guasti, L., Di Stasio, M., Miotti, B., Monteriù, A., Blikstein, P. (eds) Makers at School, Educational Robotics and Innovative Learning Environments. Lecture Notes in Networks and Systems, vol 240. Springer, Cham. https://doi.org/10.1007/978-3-030-77040-2_39 Gergen, K. J. (2001). Social construction in context. London: SAGE Publications. Gomoll, A., Hmelo-Silver, C. E., Šabanović, S., & Francisco, M. (2016). Dragons, ladybugs, and softballs: Girls’ STEM engagement with human-centered robotics. Journal of Science Education and Technology, 25(6), 899-914. doi:10.1007/s10956-016-9647-z Goodwin, K., & Highfield (2013). A framework for examining technologies and early mathematics learning. In L. D. English & J. T. Mulligan (Eds.), Reconceptualising early mathematics learning (pp. 205-226). New York: Springer. Greenberg, R. I., Thiruvathukal, G. K., & Greenberg, S. T. (2019). Integrating mathematics and educational robotics: Simple motion planning. Robotics in education (pp. 262-269). Springer International Publishing. https://doi.org/10.1007/978-3-030-26945-6_23 Gura, M. (2012). Lego robotics: Stem sport of the mind. Learning & Leading with Technology, 40(1), 12. 133 Gyebi, E. B. B., Hanheide, M., & Cielniak, G. (2017). The effectiveness of integrating educational robotic activities into higher education computer science curricula: A case study in a developing country. Educational robotics in the makers era (pp. 73-87). Springer International Publishing. https://doi.org/10.1007/978-3-319-55553-9_6 Hadwin, A. F., Järvelä, S., & Miller, M. (2018). Self-regulation, co-regulation, and shared regulation in collaborative learning environments. In D. H. Schunk, & J. A. Greene, (Eds.). Handbook of self-regulation of learning and performance (2nd ed.), pp. 83-106. New York: Routledge. Hattie, J., & Ebooks Corporation. (2012;2011;). Visible learning for teachers: Maximizing impact on learning. Routledge. https://doi.org/10.4324/9780203181522 Highfield, K. (2010). Robotic toys as a catalyst for mathematical problem solving. Australian Primary Mathematics Classroom, 15(2), 22. Highfield, K., Mulligan, J., & Hedberg, J. (2008). Early mathematics learning through exploration with programmable toys. In Proceedings of the Joint Meeting of PME (Vol. 32, pp. 169-176). Hmelo-Silver, C. E., & Barrows, H. S. (2006). Goals and strategies of a problem-based learning facilitator. Interdisciplinary Journal of Problem-Based Learning, 1(1), 4. doi:10.7771/1541-5015.1004 https://curriculum.gov.bc.ca/sites/curriculum.gov.bc.ca/files/pdf/curriculum_intro.pdf Hudson, M., & Baek, Y. (2022). Increasing elementary students' computational thinking skills using a multifaceted robotics-based intervention. Computers in the Schools, 39(1), 16-40. https://doi.org/10.1080/07380569.2022.2037295 134 James, S. M., & Singer, S. R. (2016). From the NSF: The national science foundation's investments in broadening participation in science, technology, engineering, and mathematics education through research and capacity building. CBE Life Sciences Education, 15(3), fe7-fe7. doi:10.1187/cbe.16-01-0059 Järvelä, S. & Järvenoja, H. (2006). Socially constructed self-regulated learning and motivation regulation in collaborative learning groups. Teachers College Record, 113 (2), 350-374. Järvelä, S., and Hadwin, A. F. (2013). New frontiers: regulating learning in CSCL. Educ. Psychol, 48, 25–39. doi: 10.1080/00461520.2012.748006 Järvelä, S., Järvenoja, H., & Veermans, M. (2008). Understanding the dynamics of motivation in socially shared learning. International Journal of Educational Research, 47, 122-135. Juliani, A. J. (2015). Inquiry and innovation in the classroom: Using 20% time, genius hour, and PBL to drive student success. New York, NY: Routledge. Jurado, E., Fonseca, D., Coderch, J., & Canaleta, X. (2020). Social STEAM Learning at an Early Age with Robotic Platforms: A Case Study in Four Schools in Spain. Sensors (Basel, Switzerland), 20(13), 3698. https://doi.org/10.3390/s20133698 Kelley, T. R., & Knowles, J. G. (2016). A conceptual framework for integrated STEM education. International Journal of STEM Education, 3(1), 1. https://doi.org/10.1186/s40594-016-0046-z Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75-86. Larkin, K., & Miller, J. (2020). Digital technologies and numeracy—Synergy or discord? In A. MacDonald, L. Danaia, & S. Murphy (Eds.), STEM education across the learning 135 continuum (pp. 137–154). Springer Singapore. https://doi.org/10.1007/978-981-15-2821-7_8 Lester, F. K., & Mau, S. T. (1993). Teaching mathematics via problem solving: A course for prospective elementary teachers. For the learning of mathematics, 13(2), 8-11. Li, Y., & Schoenfeld, A. H. (2019). Problematizing teaching and learning mathematics as “given” in STEM education. International Journal of STEM Education, 6(1), 1-13. https://doi.org/10.1186/s40594-019-0197-9 Liljedahl, P., Santos-Trigo, M., Bruder, R., Malaspina, U., Springer Education eBooks 2016 English/International, Open Research Library (Open Access), & SpringerLink Fully Open Access Books. (2016). Problem solving in mathematics education. Springer. https://doi.org/10.1007/978-3-319-40730-2_1 Luchins, A. S. (1942). Mechanization in problem solving-the effect of “einstellung”. Psychological Monographs, 54, 95. Mangold, J., & Robinson, S. (2013). The engineering design process as a problem solving and learning tool in K-12 classrooms. Paper presented at American Society for Engineering Education-ASEE, Atlanta, 2013. Masubuchi, Y., Sawada, H., & WaDa, I. (2022). Development of self-regulated learning abilities through the SRL cycle in junior high school science. JSSE Research Report, 36(4), 51-56. https://doi.org/10.14935/jsser.36.4_51 Matthews, M. R. (1993). Constructivism and science education: Some epistemological problems. Journal of Science Education and Technology, 2(1), 359-370. https://doi.org/10.1007/BF00694598 136 McDonald, S., & Howell, J. (2011). Watching, creating, and achieving: Creative technologies as a conduit for learning in the early years: Creative technologies in the early years. British Journal of Educational Technology, 43(4), 641-651. https://doi.org/10.1111/j.1467-8535.2011.01231.x McWhorter, W. I. (2008). The effectiveness of using LEGO® Mindstorms® robotics activities to influence self -regulated learning in a university introductory computer programming course Metallidou, P., & Vlachou, A. (2010). Children's self-regulated learning profile in language and mathematics: The role of task value beliefs. Psychology in the Schools, 47(8), 776-788. doi:10.1002/pits.20503 National Academies of Sciences, Engineering, and Medicine. 2001. Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press. https://doi.org/10.17226/9822. National Center for Education Statistics. (2008). National Center for Education Statistics - Digest of Education Statistics. Institute of Education Sciences U.S. Department of Education. National Council of Teachers of Mathematics (NCTM). (1989, 2000, 2013). Curriculum and evaluation standards for school mathematics. Reston, VA: The Council. National Council of Teachers of Mathematics. (2000a, 2000b). Executive Summary Principles and standards for school mathematics. Retrieved from https://www.nctm.org/uploadedFiles/Standards_and_Positions/PSSM_ExecutiveSummary.pdf 137 National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. Nilson, L. B. (2013). Creating self-regulated learners: Strategies to strengthen students' self-awareness and learning skills (First ed.). Stylus Publishing. Noh, J., & Lee, J. (2020). Effects of robotics programming on the computational thinking and creativity of elementary school students. Educational Technology Research and Development, 68(1), 463-484. https://doi.org/10.1007/s11423-019-09708-w Ntuli, E. (2015). Active learning strategies in technology integrated K-12 classrooms. Handbook of Research on Educational Technology Integration and Active Learning (pp. 140-162). IGI Global. Ontario Ministry of Education. (2020). The Ontario Curriculum Grades 1–8: Mathematics, 2020. Ortiz, A. M. (2015). Examining students' proportional reasoning strategy levels as evidence of the impact of an integrated LEGO robotics and mathematics learning experience. Journal of Technology Education, 26(2) doi:10.21061/jte. v26i2.a.3 Ortiz, A. M., Bos, B., & Smith, S. (2015). The power of educational robotics as an integrated STEM learning experience in teacher preparation programs. Journal of College Science Teaching, 44(5), 42. Papadakis, S., Vaiopoulou, J., Sifaki, E., Stamovlasis, D., & Kalogiannakis, M. (2021). Attitudes towards the use of educational robotics: Exploring pre-service and in-service early childhood teacher profiles. Education Sciences, 11(5), 204. https://doi.org/10.3390/educsci11050204 138 Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books. Papert, S. (1993). The children's machine: Rethinking school in the age of the computer. New York: BasicBooks. Papert, S. (1999). Introduction: What is LOGO and who needs it? In Logo Philosophy and Implementation (pp. v-xvi). Retrieved from http://www.microworlds.com/company/philosophy.pdf Papert, S., & Harel, J. (1991). Situating constructionism. Constructionism, 36, pp. 1–11. Pérez-González, J., Filella, G., Soldevila, A., Faiad, Y., & Sanchez-Ruiz, M. (2022). Integrating self-regulated learning and individual differences in the prediction of university academic achievement across a three-year-long degree. Metacognition and Learning, 17(3), 1141-1165. https://doi.org/10.1007/s11409-022-09315-w Perry, N. E. (1998). Young children's self-regulated learning and contexts that support it. Journal of Educational Psychology, 90(4), 715-729. doi:10.1037/0022-0663.90.4.715 Perry, N., Phillips, L., & Hutchinson, L. (2006). Mentoring student teachers to support self‐regulated learning. The Elementary School Journal, 106(3), 237-254. doi:10.1086/501485 Peters, K. (2007). m-Learning: Positioning educators for a mobile, connected future. International Journal of Research in Open and Distance Learning, 8(2), 1-17. Piaget, J. (1973). To understand is to invent. New York: Grossman. Pintrich, P. R., and de Groot, E. V. (1990). Motivational and self-regulated learning components of classroom academic performance. J. Educ. Psychol. 82, 33–40. doi: 10.1037/0022-0663.82.1.33 139 Pintrich, P. R., Marx, R. W., and Boyle, R. A. (1993a). Beyond cold conceptual change: the role of motivational beliefs and classroom contextual factors in the process of conceptual change. Review of Educational Research, 63(2), 167-199. doi: 10.3102/00346543063002167 Pintrich, P. R., Smith, D. A. F., Garcia, T., and Mckeachie, W. J. (1993b). Reliability and predictive validity of the motivated strategies for learning questionnaire (MSLQ). Educational and Psychological Measurement, 53, 801–813. doi: 10.1177/0013164493053003024 Pólya, G. (1971). How to solve it: A new aspect of mathematical method (2d - ed.). Princeton, N.J: Princeton University Press. Pólya, G. (1981). Mathematical discovery: On understanding, learning, and teaching problem solving (Combin ed.). New York: Wiley. Rakes, C. R., Ronau, R. N., Bush, S. B., Driskell, S. O., Niess, M. L., & Pugalee, D. K. (2020). Mathematics achievement and orientation: A systematic review and meta-analysis of education technology. Educational Research Review, 31, 100337. https://doi.org/10.1016/j.edurev.2020.100337 Resnick, M. (2017). Lifelong Kindergarten: Cultivating Creativity through Projects, Passions, Peers, and Play. MIT Press. Rigelman, N. R. (2007). Fostering mathematical thinking and problem solving: The teacher's role. Teaching Children Mathematics, 13(6), 308-314. https://doi.org/10.5951/TCM.13.6.0308 140 Ryan, A. M., & Pintrich, P. R. (1997). “Should I ask for help?” The role of motivation and attitudes in adolescents' help seeking in math class. Journal of Educational Psychology, 89(2), 329–341. https://doi.org/10.1037/0022-0663.89.2.329 Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children (1st ed.). New York: Routledge. Savard, A., & Highfield, K. (2015). Teachers’ talk about Robotics: Where is the Mathematics? In M. Marshman, V. Geiger, & A. Bennison (Eds.). Mathematics education in the margins (Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia), pp. 540–546. Sunshine Coast: MERGA Scaradozzi, D., Screpanti, L., & Cesaretti, L. (2019). Towards a definition of educational robotics: A classification of tools, experiences and assessments. Smart learning with educational robotics (pp. 63-92). Springer International Publishing. https://doi.org/10.1007/978-3-030-19913-5_3 Scheonfeld, A. H. (1988) When Good Teaching Leads to Bad Results: The Disasters of “Well Taught” Mathematics Courses. Berkeley, CA. Schoenfeld, A. H. (1982). Some thoughts on problem-solving research and mathematics education. In F. K. Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in research (pp. 27–37). Philadelphia: Franklin Institute Press. Schoenfeld, A. H. (1985a). Making sense of “out loud” problem-solving protocols. The Journal of Mathematical Behavior, 4(2), 171-191. Schoenfeld, A. H. (1988). Problem solving in context(s). In R.I. Charles & E.A.Silver (Eds.), The teaching and assessing of mathematical problem solving (Research Agenda for 141 Mathematics Education, vol. 3, pp. 82–92). Reston, VA: National Council of Teachers of Mathematics. Schoenfeld, A. H. (1989). Explorations of students’ mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20, 338–355. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan. Schoenfeld, A. H. (2006). What doesn’t work: The challenge and failure of what works clearinghouse to conduct meaningful reviews of studies of mathematics curricula. Educational Researcher, 35(2), 13–21 Schoenfeld, A. H. (2010;2011). How we think: A theory of goal-oriented decision making and its educational applications. Routledge. https://doi.org/10.4324/9780203843000 Schoenfeld, A. H. (2016). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics (reprint). Journal Of Education,196(2), 1-38. Schoenfeld, A.H. (2022). Why are learning and teaching mathematics so difficult?. In: Danesi, M. (eds) Handbook of Cognitive Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-44982-7_10-1 Schroeder, T. L., & Lester, F. K., Jr. (1989). Developing understanding in mathematics via problem solving. In P. R. Trafton (Ed.), New directions for elementary school mathematics, 1989 Yearbook of the National Council of Teachers of Mathematics (pp. 31-42). Reston, VA: National Council of Teachers of Mathematics. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19-28. 142 Song, H., Tsiakas, K., Ham, J., Markopoulos, P., & Barakova, E. I. (2024). ‘How would you score yourself?’: The effect of self-assessment strategy through robots on Children’s motivation and performance in piano practice. International Journal of Social Robotics, 16(2), 327-340. https://doi.org/10.1007/s12369-023-01080-3 Stevenson, H. W., & Stigler, J. W. (1992). The learning gap: Why our schools are failing and what we can learn from Japanese and Chinese education. Summit Books. Stokes, A., Aurini, J., Rizk, J., Gorbet, R., & McLevey, J. (2022). Using robotics to support the acquisition of STEM and 21st-century competencies: Promising (and practical) directions. Canadian Journal of Education, 45(4), 1141-1170. https://doi.org/10.53967/cje-rce.5455 Su, J., Yang, W., & Zhong, Y. (2023). Influences of gender and socioeconomic status on children's use of robotics in early childhood education: A systematic review. Early Education and Development, 34(4), 910-926. https://doi.org/10.1080/10409289.2022.2078617 Sullivan, A., & Bers, M. U. (2016). Robotics in the early childhood classroom: Learning outcomes from an 8-week robotics curriculum in pre-kindergarten through second grade. International Journal of Technology and Design Education, 26(1), 3-20. https://doi.org/10.1007/s10798-015-9304-5 Sullivan, A., & Bers, M. U. (2019). Investigating the use of robotics to increase girls’ interest in engineering during early elementary school. International Journal of Technology and Design Education, 29(5), 1033-1051. https://doi.org/10.1007/s10798-018-9483-y Sung, W., Ahn, J., Ming Kai, S., Choi, A., & Black, J. B. (2016). Incorporating touch-based tablets into classroom activities: Fostering children’s computational thinking through 143 iPad integrated instruction. In D. Mentor (Ed.), Handbook of research on mobile learning in contemporary classrooms (pp. 378–406). IGI Global. https://doi.org/10.4018/978-1-5225-0251-7.ch019 Szabo, Z. K., Körtesi, P., Guncaga, J., Szabo, D., & Neag, R. (2020). Examples of problem-solving strategies in mathematics education supporting the sustainability of 21st-century skills. Sustainability (Basel, Switzerland), 12(23), 10113. https://doi.org/10.3390/su122310113 Tee, K. N., Leong, K. E., & Abdul Rahim, S. S. (2021). A self-regulation model of mathematics achievement for eleventh-grade students. International Journal of Science and Mathematics Education, 19(3), 619-637. https://doi.org/10.1007/s10763-020-10076-8 Tripathi, P. N. (2008). Developing mathematical understanding through multiple representations. Mathematics Teaching in the Middle School, 13(8), 438-445. https://doi.org/10.5951/MTMS.13.8.0438 Tzohar-Rozen, M., & Kramarski, B. (2014). Metacognition, motivation and emotions: Contribution of self-regulated learning to solving mathematical problems. Global Education Review, 1(4), 76-95. University of York. (2023). Home page, University of York, https://online.york.ac.uk/what-is-computational-thinking/ Van Meeteren, B. & Zan, B. (2010). Revealing the work of young engineers in early childhood education. STEM in Early Education and Development Conference. Vassallo, S. (2013). Self-regulated learning: An application of critical educational psychology. New York: Peter Lang. 144 Wang, Y., & Sperling, R. A. (2020). Characteristics of effective self-regulated learning interventions in mathematics classrooms: A systematic review. Frontiers in Education, 5(58). https://doi.org/10.3389/feduc.2020.00058 Williams, K., Igel, I., Poveda, R., Kapila, V., & Iskander, M. (2012). Enriching K-12 science and mathematics education using LEGOs. Advances in Engineering Education, 3(2) Wilson, J. W., Fernandez, M. L., & Hadaway, N. (1993). Mathematical problem solving. In P. S. Wilson (Ed.), Research ideas for the classroom: Highschool mathematics (pp. 57–77). NCTM. Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49, 33–35. Wing, J. M. (2008). Computational thinking and thinking about computing. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical, and Engineering Sciences, 366(1881), 3717-3725. https://doi.org/10.1098/rsta.2008.0118 Winne, P. H., and Hadwin, A. F. (1998). “Studying as self-regulated engagement in learning,” in Metacognition in Educational Theory and Practice, eds D. Hacker, J. Dunlosky, and A. Graesser (Hillsdale, NJ: Erlbaum), 277–304. Winne, P. H., and Hadwin, A. F. (2008). “The weave of motivation and self-regulated learning,” in Motivation and Self-Regulated Learning: Theory, Research and Applications, eds D. H. Schunk and B. J. Zimmerman (New York, NY: Lawrence Erlbaum Associates), 297–314. Winne, P. H., and Hadwin, A. F. (2013). “Study: tracing and supporting self-regulated learning in the internet,” in International Handbook of Metacognition and Learning Technologies, eds R. Azevedo and V. Aleven (New York, NY: Springer), 293–308. 145 Woods, D.R., Wright, J.D., Hoffman, T.W., Swartman, R.K., Doig, I.D. (1975). Teaching Problem solving Skills. Yannier, N., Hudson, S. E., Koedinger, K. R., Hirsh-Pasek, K., Golinkoff, R. M., Munakata, Y., Doebel, S., Schwartz, D. L., Deslauriers, L., McCarty, L., Callaghan, K., Theobald, E. J., Freeman, S., Cooper, K. M., & Brownell, S. E. (2021). Active learning: “hands-on” meets “minds-on”. Science (American Association for the Advancement of Science), 374(6563), 26-30. https://doi.org/10.1126/science.abj9957 Yin, R. K. (2014). Case study research: Design and methods (5th ed.). Los Angeles: Sage. Zimmerman, B. J. (2000). Attainment of self-regulated: A social cognitive perspective. In M. Boekaerts, P. Pintrich, & M. Zeidner (Eds.), Handbook of self-regulation: Research and applications (pp. 13–39). Orlando: Academic Press. Zimmerman, B. J. (2001). Theories of self-regulated learning and academic achievement: An overview and analysis. In B. J. Zimmerman & D. H. Schunk (Eds.), Self-regulated learning and academic achievement: Theoretical perspectives (2nd ed., pp. 1-38). Mahwah, NJ: Lawrence Erlbaum Associates. 33. Zimmerman, B. J., & Kitsantas, A. (2014). Comparing students’ self-discipline and self-regulation measures and their prediction of academic achievement. Contemporary Educational Psychology, 39(2), 145-155. https://doi.org/10.1016/j.cedpsych.2014.03.004 Zimmerman, B. J., & Martinez-Pons, M. (1988). Construct validation of a strategy model of student self-regulated learning. Journal of Educational Psychology, 80(3), 284-290. Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts 146 Appendices Appendix A: Sample Problem Solving Activities Screenshots A.1 Windmill (clock option) Screenshot A.2 Butterfly Project Screenshot 147 Appendix B: Coding Categories Instrument SRL Examples from field notes Setting goals “We are making a robot that can pick up objects and place them there”. Planning Not observed. Enacting Strategies “Let's make it faster.” Monitoring “That is better”, the student keeps an eye on the robot arm to avoid any unexpected obstacles. Elaborating Students search online for pictures of construction. Collaborating “I will do the code now. How about you continue to build?” Adjusting “Let's change the power to 1.” Persisting Fix a broken robot. Reflection Not observed Mathematics Examples from field notes Reading Students read out aloud the problem description on the screen. Estimating “It looks like a 9 [hole beam].” Computing “8+8 is 16. 16+8 is 24” Experimenting Students try different inputs in coding Checking “Is this good?” Mathematics Examples from field notes Modelling Not observed Symbolism Students write equations on paper. Note. Items of SRL are adapted and modified from Perry, N. E. (1998). Young children's self-regulated learning and contexts that support it. Journal of Educational Psychology, 90(4), 715-729. Items of Mathematics are adapted and modified from Schoenfeld, A. H. (2011;2010). How we think: A theory of goal-oriented decision making and its educational applications. Routledge. https://doi.org/10.4324/9780203843000 148 Appendix C: Field Notes 149 Appendix D: Interview Questions 1. Did you understand the task? What was it about? - This question aimed to encourage them to articulate their understanding and provide context to their subsequent actions. 2. What was your idea to do it? Did it work? - This inquiry prompted them to articulate their initial approach and allowed for an evaluation of the effectiveness of their chosen strategies. 3. Have you had some difficulties? - This question aimed to uncover challenges encountered during the robotics activity, fostering a discussion, and prompting the students to reflect on obstacles and errors they may have faced. 4. Have you used math? - Exploring the integration of mathematical concepts within the robotics tasks, this question sought to understand the extent to which participants applied mathematical principles in their problem-solving processes. 5. Do you think you did a good job? - This reflective query invited participants to evaluate their own performance, providing valuable insights into their perceptions of their achievements. 150 Appendix E: Interview Excerpt Subcase:1 Site: C Mar.21,2019 4:40pm 1. Interviewer: What's our goal today? 2. Sylvia: Hammer. Make the hammer smash the ghost… yeah, it's an invisible ghost... 3. Interviewer: When you heard the problem, what's your plan? You want to rush it to the end? 4. Sylvia: No, I just take my time and do it. 5. Interviewer: Do you feel that it takes too long today? 6. Sylvia: Yeah, [the problem] is just a little hard. 7. Interviewer: Did you worry about your plan not working? 8. Sylvia: No. 9. Interviewer: How so? 10. Sylvia: I just focus. 11. Interviewer: How did your plan work out today? 12. Sylvia: There are good and bad. 13. Interviewer: Can you talk about the good part? 14. Sylvia: Some of it eventually worked. 15. Interviewer: And the bad part? 16. Sylvia: Like I kept on having these problems. 17. Interviewer: Which problems? 18. Sylvia: Like it wouldn't work, or the black piece fell out. 19. Interviewer: What are you doing over there? 20. Sylvia: Counting. It's two and I'm goanna add how many twos. I added two on there. Code: Persistence Code: Persistence Code: Errors 151 21. Interviewer: You automatically counted by two? 22. Sylvia: Yeah, I don't know why. I counted by two. 23. Interviewer: What do you think about your strengths and weaknesses [in math]? 24. Sylvia: Um… my weakness is my mind always goes to two counting by twos… um… strength I can count by all kinds of numbers. 25. Interviewer: How do you like your plan? 26. Sylvia: My plan was good. I came up with one and then copy it [she meant she came up with the equation on paper first and then translated it into coding]. 27. Interviewer: The second one didn't work out. 28. Sylvia: The first one did. 29. Interviewer: I told you there is more than one solution, right? 30. Sylvia: Yeah. 31. Interviewer: You're okay with only one as long as you have one. 32. Sylvia: I don't. 33. Interviewer: What if the teacher gives you only partial marks for this question because you didn't get all the solutions? Do you care? 34. Sylvia: I don't. 35. Interviewer: Why is that? Many kids might care. 36. Sylvia: I don't have grades in school.
Thesis/Dissertation
eng
Curriculum Studies
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Anderson, Ann
Exploring children’s mathematics learning and self-regulation in robotics
Text