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Felix Klein's notion of "double discontinuity" between university mathematics and secondary school mathematics has persisted in mathematics teacher education. Situating this study in Hong Kong, we investigated three prospective secondary teachers' experiences in higher education and secondary mathematics classrooms, including their figured worlds, identities, and mathematical knowledge constructed through those experiences. Our analysis revealed their contrasting experiences in these two contexts, adding to our understanding of the phenomenon of "double discontinuity" in terms of not only how it is relevant to Hong Kong but also how it may manifest in the specific cultural context. Moreover, we found coherency and continuity during the prospective teachers' secondary-tertiary transition; their learning experiences in both environments served as critical resources upon which they reflected to envision their future mathematics teaching. We conclude with implications and suggestions for mathematics teacher education to support prospective teachers' (and their future students') transitions between high school and university mathematics.

Grounded in problem-based learning and with respect to four mathematics domains (arithmetic, random events and counting, number theory, and geometry), we designed a series of programming-based learning tasks for middle school students to co-develop computational thinking (CT) and corresponding mathematical thinking. Various CT concepts and practices articulating the designated mathematical problems were involved in the tasks. In addition to delineating the design of these learning tasks, this paper presents a qualitative study in which we examined 74 students' learning outcomes and characterized their CT and mathematical thinking co-development as they accomplished the tasks. The research results demonstrate the codevelopment of both mathematics- and CT-related concepts and practices in the four mathematics domains. Two types of interactions are identified: (i) applying mathematical knowledge to construct CT artifacts and (ii) generating new mathematical knowledge with CT practice. The new insights provided by the present work are threefold. First, from a mathematical learning perspective, the nature of the solution processes of the designed problems should not be immediately obvious. Second, from a technology-enhanced learning perspective, the dynamic representations and immediate visual feedback afforded by the programming tool are beneficial to student learning. Third, from a pedagogical perspective, the room for customization offered by both the designed problems and programming tools can provide affordances for learning.

Creativity, one of the cornerstones of students' 21st-century skills, is regarded as an important learning outcome of science, technology, engineering, arts, and mathematics (STEAM) education. Meanwhile, problem-based digital making (DM), which combines the child-friendly programming activities of DM with problem-solving elements, is an emerging instructional design to facilitate STEAM learning. This qualitative case study examines the implementation of a problem-based DM instructional program that used the block-based programming tool Scratch to cultivate the participants' creativity. Fifty-four middle school students (aged 10-14 years) in Hong Kong participated in the program, which totaled 10 contact hours over five consecutive weeks. Through triangulating students' DM artifacts, video recordings, field notes, and interviews, the researchers characterized the students' creative expression, examined the role of problem-based DM in encouraging creative work, and investigated the use of Scratch for mediating student creativity. The results showed that problem-based DM activities fostered students' creative expressions in the dimensions of novelty, utility, aesthetics, and authenticity. While Scratch mediated the way the students presented their solutions, it had limitations that hindered the students' digital artifact construction. The findings provide theoretical insights for framing creativity and offer practical implications for the implementation of problem-based DM in K-12 contexts.

There has been substantial research undertaken on the integration of computational thinking (CT) in K-12 mathematics education in recent years, particularly since 2018 when relevant systematic reviews were conducted on the topic. Many empirical studies in this area have yet to elaborate clearly and explicitly on how CT may support mathematics learning, or otherwise, in CT-based mathematics activities. Addressing this research gap, we conducted a systematic review on the integration of CT in K-12 mathematics education with a focus on CT-based mathematics instruction and students learning under such instruction. The Web of Science database was searched for in terms of studies published from 2006 to 2021, from which 24 articles were selected to provide illustrations of CT-based mathematics instruction and related student learning, and they were further analyzed according to education levels and contexts, programming tools, learning outcomes in CT and mathematics, and the mutual relationship between CT and mathematics learning. Among the results, this review found that geometrized programming and student-centered instructional approaches were facilitators of productive learning in CT and mathematics. Moreover, CT-based mathematics learning entails an interactive and cyclical process of reasoning mathematically and reasoning computationally, which can occur when: (1) applying mathematics to construct CT artefacts; (2) applying mathematics to anticipate and interpret CT outputs; and (3) generating new mathematical knowledge in parallel with the development of CT. The findings contribute to an in-depth understanding of what, and how, CT-based mathematics instruction impacts student learning in K-12 contexts.

We report on a case study of eight grade 5 to 6 students (ages 12-14) involved in a three-day "digital-making summer camp" focused on mathematical problem-solving through block-based programming combined with programmable electronics. Data analysis focused on the computational thinking (CT), mathematics, and problem-solving challenges that surfaced during the activities and on the students' developing perspectives about themselves as computational thinkers and problem solvers. Our results suggest the students developed various CT and problem-solving competences and dispositions during the digital-making activities. However, they experienced difficulties with ill-structured problems. This study supports the creative uses of digital making as complementary to K-12 mathematics learning.

Programming is an interdisciplinary practice with applications in both mathematics and computer science. Mathematics concerns rigor, abstraction, and generalization. Computer science predominantly concerns efficiency, concreteness, and physicality. This makes programming a medium for problem solving that mediates between mathematics and computer science in intriguing ways. Behind programming practices is computational thinking (CT), a mode of thinking involved in formulating and solving problems so that the solutions could be represented and carried out by computing means. In this paper, CT is seen as a boundary object connecting mathematics and computer science in a school problem-solving context. In particular, we examine and analyse middle school students' work upon engaging in mathematical problem solving-in a programming environment, taking CT as a boundary object embedded in the block-based programming environment, Scratch. The analysis is guided by observing boundary crossing features of CT in the students' artefacts produced in Scratch while solving mathematical problems related to symmetry and arithmetic sequence. The findings of this study open up new dimensions to explore CT as a boundary object in integrated STEM pedagogy.

Amid the maker movement, educators are proposing various making activities with programmable artifacts to prepare students for coping with the challenges in the twenty-first century. Today, the "4C" skills--critical thinking, creativity, communication, and collaboration--are regarded as significant learning outcomes in Science, Technology, Engineering, and Mathematics education; however, few researchers have investigated the adoption of problem-based learning in K-12 programming education for developing students' 4C skills. A case study was conducted in a "digital making" camp in which 54 upper elementary and lower secondary school students (10-14 years old) were engaged in harnessing a block-based programming tool, Scratch, to conduct various problem-solving tasks. Through triangulating multiple sources of qualitative data (including lesson plans, classroom field notes, videotaped lesson records, student solutions/artifacts, and post-intervention interviews), together with the microgenetic learning analysis, this study characterizes students' 4C skills development in the process of problem-based digital making. We found that the problem-based digital making environment supported the students' development of: (a) critical thinking in the form of critical modeling and critical data handling; (b) creativity in the form of creative explorations, creative solutions, and creative expressions; and (c) communication and collaboration in the form of communicative scaffolding and collaborative debugging. Complementary evidence-based suggestions for scaffolding problem-based digital making activities are suggested.

This paper reports on a mixed-methods, design-based study conducted with five classes of Primary 6 (age 11-12 years) students learning the properties of 3D solids. The researcher and participating teachers designed mathematics lessons that integrated the hands-on construction of artifacts (i.e., making) with a class set of 3D printing pens to introduce various properties of prisms and pyramids in two primary schools in Hong Kong. Quantitative results show that the designed learning activities with 3D printing technology greatly supported student learning on all questions related to working with vertices (0D), edges (1D), and faces (2D). Qualitatively, one researcher observed that the students produced hand movements and language that were conducive to learning the properties of 3D solids while they interacted with 3D printing pens. This study draws attention to the unity of mind-body and body-tool interactions in the act of making something (i.e., mathematics learning as embodied making).

We report on an action research study conducted by two mathematics teacher educators in the United Kingdom and Hong Kong. The project consisted of three phases, the first two taking place in our respective teacher education courses and a final phase in which prospective mathematics teachers from both locations engaged in dialogue through a speculative fiction task. Our aim was to explore how our prospective mathematics teachers can be supported in becoming mathematics teachers for sustainable futures by innovating our own mathematics teacher education curricula. This study suggests that generative methods which support teachers in speculating on paths to potential futures as teachers of mathematics, can provide access to how prospective teachers negotiate issues in relation to the climate crisis and ways in which they conceive of related changes in their practices. By identifying the aspects of mathematics teaching that are viewed by the prospective teachers as contingent, and through offering tasks that support an expansion of this view, teacher educators are better placed to support the development of teaching practices needed for sustainable futures.

In this paper, we introduce a technology-enhanced pedagogical sequence for supporting lower secondary school students' sense-making of the concept of volume in a non-procedural and non-formula-driven way. Specifically, we illustrate a novel approach of using dynamic geometric environment (DGE) to introduce the meaning of volume and then deriving the volume of a hemisphere with minimal algebra but simple trigonometry. This method aims to support students' geometrical reasoning through comparing the volume of different three-dimensional shapes and analysing their cross sections two-dimensionally, thus drawing on the concept of area. We highlight the affordances of the designed DGE sketches for fostering a dynamic conception of volume.

This study examines how teachers' discourse moves influence patterns of authority in the mathematics classroom. We identify changes in authority structure in one classroom taught by a primary mathematics teacher in Hong Kong. This teacher participant was one of 32 who participated in a larger teacher-intervention study that aimed to help teachers orchestrate dialogic classroom discourse for rich learning opportunities in mathematics. Previous research has investigated teachers' discourse behaviors and authority structures in mathematics classrooms, but the connection between the two has yet to be addressed in extant studies. Our analysis shows that certain features and patterns embedded in classroom discourse can constrain or enable certain kinds of authority in the classroom. This study contributes to understanding the ways in which dialogic discourse may influence how authority is structured, including how mathematics teaching and learning are enacted through authority styles, and how the discipline of mathematics is positioned from an authority perspective.

This paper reports on the design, implementation, and results of a video-based study that focused on supporting an individualized viewing experience, keeping a record of what was noticed, providing a degree of guiding framework, and facilitating a combination of individual and collaborative reflections in noticing activities for STEM professional development. A cohort of 20 prospective mathematics teachers at a public university in Hong Kong participated in the study. The participants engaged in a series of blended learning activities with a feature that allowed them to leave comments while viewing of a lesson video, which focused on technology integration in mathematics classrooms. The designed activities were helpful for the participants to document and reflect on their observations, while viewing videos outside the classroom allowed them to meaningfully engage with the course materials. Moreover, the use of videos supported the participants' broad but crucial STEM classroom practices that would help them prepare for future lessons.

In this paper, we explore the challenges experienced by a group of Primary 5 to 6 (age 12-14) students as they engaged in a series of problem-solving tasks through block-based programming. The challenges were analysed according to a taxonomy focusing on the presence of computational thinking (CT) elements in mathematics contexts: preparing problems, programming, create computational abstractions, as well as troubleshooting and debugging. Our results suggested that the challenges experienced by students were compounded by both having to learn the CT-based environment as well as to apply mathematical concepts and problem solving in that environment. Possible explanations for the observed challenges stemming from differences between CT and mathematical thinking are discussed in detail, along with suggestions towards improving the effectiveness of integrating CT into mathematics learning. This study provides evidence-based directions towards enriching mathematics education with computation.

This paper reports on a design-based study within the context of a 3-day "digital making" (DM) summer camp attended by a group of students (aged 11-13) in grades 5 and 6. During the camp, students were presented with a set of mathematical problems to solve in a block-based programming environment, which was connected to various physical input sensors and output devices (e.g., push buttons, LED lights, number displays, etc.). Students' code files, and screen captures of their computer work, were analyzed in terms of their developed computational problem-solving practices and any computational concepts that emerged during the problem-based DM. The results suggested that the designed tasks consistently supported the students' modeling and algorithmic thinking, while also occasioning their testing and debugging practices; moreover, the students utilized computational abstractions in the form of variables, and employed different approaches, to formulate mathematical models in a programming context. This study contributes to the 'big picture' of how using computers might fundamentally change mathematics learning, with an emphasis on mathematical problem-solving. It also provides empirically grounded evidence to enhance the potential of computational thinking as a new literacy, and problem-solving as a global competence, in formal school settings.

We adopt five observation categories, namely classroom management, classroom environment, communication, mathematical content and tasks, to analyse four in-service secondary mathematics teachers' noticing upon watching video episodes showing an actual mathematics lesson that implemented 3D Printing Pens for teaching and learning shape and space. We use coding to analyse what the participants generally identified as important or noteworthy in the video. Moreover, we employ thematic analysis to delve deeper into the participants' interpretations and decisions in relation to using 3D Pens for teaching and learning mathematics. Our findings have implications for teachers' professional development in the area of technology integration especially in terms of their realisation of the affordances of novel-to-them technologies. We also report methodological and conceptual contributions towards teacher noticing.

In this paper, I discuss the potential transformations to teach K-12 geometry topics with the technology of 3D Pens, which enabled 3D models to be created instantly via one's moving hands. I draw on an example of using constructionist learning activities with 3D Pens to teach young children, 'how tall is the triangle?'

This paper documents the process and outcome of the development and validation of a coding scheme to analyse mathematics classroom discourse in the context of Hong Kong primary schools. The objective of this coding scheme was to systematically analyse teacher discourse behaviours before and after an intervention programme; the programme aimed to assist in-service teachers to develop conceptual and practical tools for engaging students in dialogical discourse in mathematics classrooms. This paper provides a detailed outline of what took place in designing and refining the coding scheme as well as improving its validity and reliability. The developed coding scheme can be effective for highlighting dialogic discourse, authority structures and student participation in mathematics classrooms and beyond. Consequently, it was possible to assess the efficacy of the teacher intervention programme. On the other hand, the developed coding scheme could be used to help teachers to gain a sense of efficacy over their instruction by giving them a roadmap for what they can do before and during instruction.

A growing body of research is focusing on integrating artificial intelligence (AI) and computational thinking (CT) to enhance student learning outcomes. Many researchers have designed instructional activities to achieve various learning goals within this field. Despite the prevalence of studies focusing on instructional design and student learning outcomes, how instructional efforts result in the integration of AI and CT in students' learning processes remains unclear. To address this research gap, we conducted a systematic literature review of empirical studies that have integrated AI and CT for student development. We collected 18 papers from four prominent research databases in the fields of education and AI technology: Web of Science, Scopus, IEEE, and ACM. We coded the collected studies, highlighting the students' learning processes in terms of research methodology and context, learning tools and instructional design, student learning outcomes, and the interaction between AI and CT. The integration of AI and CT occurs in two ways: the integration of disciplinary knowledge and leveraging AI tools to learn CT. Specifically, we discovered that AI- and CT-related tools, projects, and problems facilitated student-centered instructional designs, resulting in productive AI and CT learning outcomes. [ABSTRACT FROM AUTHOR]

It has been well documented that active learning (AL) improves student learning outcomes in education. This quasi-experimental study explores the effect of active learning on students' knowledge of calculus concepts, in the form of the Calculus Concept Inventory (CCI), regular assignment scores, and test scores, during an 8-week calculus tutorial program. The active pedagogies used were cooperative problem-based learning and peer assessment, implemented using an interactive online whiteboard (IOWB) called RealtimeBoard. This study reveals statistically significant evidence that active learning increases students' conceptual understanding and graded assignment performance in a first-year calculus class in Hong Kong. Furthermore, this study contributes towards a better understanding of how active learning can be implemented effectively in Asian tertiary mathematics education.

In this paper, we build on a previously developed notion of 'learning as Making' to examine mathematics thinking and learning in a highly transformative and technological Making environment: one that involves a handheld 3D printing technology which enables 3D models to be created instantly via one's moving hand. In particular, we present two examples of Maker-centred lessons for teaching and learning of primary mathematics. In these lessons, the students actively constructed artefacts with 3D Printing Pens while engaging in inquiry-based learning activities, where the target concepts were properties of prisms and cross-sections at the primary 5 (age 10-11) and primary 6 (age 11-12) levels respectively. We use diffractive analysis to capture the fine details in students' body-material interactions while engaging in the tasks with or without the 3D Printing Pens during the lessons. Through the lens of Making as a material act of creation and seeking to update Papert's constructionist view of learning, we propose to rethink Making in school mathematics according to a four-fold characterisation: "Making is co-constructing meanings," "Making is mathematising," "Making is assembling with technology" and "Making is inventing." We discuss our contribution towards advancing a materialist perspective of learning mathematics and implications for a 'learning as Making' pedagogy and curriculum.

Background: This paper compares the effects of two classroom-based technology-enhanced teaching interventions, conducted in two schools in sixth (age 11-12) grade. In one school, the intervention involves the use of a class set of 3D Printing Pens, and in another school the use of dynamic geometry environments, for inquiry-based learning of the relations among the number of vertices, edges, and faces of prisms and pyramids. An instrument was designed as guided by the van Hiele model of geometric thinking and administered to the two groups in the form of pretests, posttests, and delayed posttests to assess students' prior knowledge before the intervention started, the learning outcomes obtained immediately after intervention, and the retention of knowledge after the interventions had been completed for a sustained period of time. The purpose of this study is to explore differences in geometry learning outcomes in two technology-enhanced environments, one that involves dynamic, visual representations of geometry and another that involves embodied actions of constructing physical 3D solids. Results: The results show that students using dynamic geometry improved at a higher rate than those using 3D Pens. On the other hand, students with the aid of 3D Pens demonstrated better retention of the properties of 3D solids than their dynamic geometry counterparts. Namely, the posttest results show that the dynamic geometry environment (DGE) group generally outperformed the 3D Pen group across categories. The observed outperformance by the DGE group on "advanced" implies that the DGE technology had a stronger effect on higher levels of geometric learning. However, the results from the ANCOVA suggest that the retention effect was more significant with 3D Pens. Conclusions: This study has established evidence that the DGE instructions produced strong but relatively temporary geometry learning outcomes, while 3D Pen instructions can help solidify that knowledge. The results of this study further shed light on the effect of visual and sensory-motor experiences on school mathematics learning and corroborate previous work showing that the effects of gesture are particularly good at promoting long-lasting learning.

In this paper, I introduce Sfard's discursive framework to examine secondary-school calculus students' communication during exploratory activities mediated by the use of touchscreen dynamic geometry environments (DGEs). Six pairs of secondary-school students participated in an open-ended task to explore calculus relationships using touchscreen-DGEs. Qualitative data capturing the students' linguistic communication (speech) and hand movements (gestures and dragging) were analysed when the students interacted with the touchscreen-DGEs used during the task. Findings suggest that new forms of communication were mobilised through the act of dragging on a touchscreen-DGE. In particular, routines for comparing, reasoning, conjecturing and verifying emerged within the use of the haptic DGE interface. In this paper, potentials of the touchscreen-DGEs in facilitating new forms of gestural thinking, as well as theoretical and methodological considerations that recognise the changing ways in which the hand and media interact, are discussed.

The recent and popular "Maker" movement worldwide has revived conversations about creativity, hands-on "Making," arts and design, humans with tools and digital experiences beyond the flat-screen. However, such conversation mainly revolved outside the realm of formal education. This article presents two learning tasks, one in upper primary (age 10-11) and one in lower secondary (age 12-13), which integrate "Making" and 3D computer-aided design (CAD), thereby facilitating Science, Technology, Engineering, and Mathematics (STEM) learning in mathematics classrooms. By adopting a design-based research methodology, we examine students' mathematics learning with respect to dissecting and forming 2D shapes and volume of composite solids with the use of 3D CAD and the kinds of integrated STEM learning practices they demonstrated in the activities. Qualitative data were collected in the form of videos of the students' communication and screenshots of their initial and final designs as they engaged with the 3D CAD environment, as well as students' written reflections and teachers' lesson analysis. Results showed that the students used 3D CAD to develop spatial skills and to achieve mathematics learning far beyond using formulae and performing procedures. The learning activities also enabled an integrated STEM learning experience in productive and unobtrusive ways.