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Abstract: Abstract The growing use of programming in mathematics classrooms presents a challenge linked to implementation in general and task design in particular. This article presents design ideas for mathematical problems incorporating programming in which the focus remains mainly on learning mathematics and less on learning programming. The article starts by reviewing the theoretical background for technology implementation and design, and then presents the methodology for the design, before exploring and discussing the design ideas with an in-depth example. Building on the idea of adidactical situations from the theory of didactical situations, the design illustrates a possible way of implementing programming in the mathematics classroom to facilitate mathematical learning. PubDate: 2024-05-06

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Abstract: In addressing the challenge of fostering functional thinking (FT) in secondary school students, our research centered on the question of how an embodied design can enhance FT’s different aspects, including input–output, covariation, and correspondence views. Drawing from embodied cognition theory and focusing on an action- and perception-based task design that uses light ray contexts and different function representations, we developed a digital-embodied learning environment, using the nomogram as a central representation. Our pilot study involving four eighth-grade students provided insights into their physical interactions with these modules through a multi-touch digital interface. Analysis of video and audio recordings from the pilots, including students’ hand gestures and verbal expressions, was guided by comparing hypothetical learning activities with the actual learning activities. The results show that (1) a concrete light ray context enables students to ground the abstract mathematical function concept; (2) the bimanual coordinating motion tasks, incorporating the covariation aspect of FT, allow students to connect their bodily experience with function properties; and (3) our embodied and dragging tasks support insight in the conversion between nomograms and graphs of functions, encouraging students’ correspondence thinking by providing multiple perspectives to understand, reason about, and manipulate the function. In conclusion, our findings suggest the potential of digital-embodied tasks in fostering FT, evident in students’ diverse strategies and reasoning. PubDate: 2024-05-03

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Abstract: Abstract Nowadays, mathematics teachers in K–12 strive to promote their students‘ mathematical knowledge and computational thinking (CT) skills. There is an increasing need for effective CT-embedded mathematics learning material and a better understanding of students’ views toward them. In this work, we present the results of a research study, which included the design of a six-lesson learning activity aimed at fostering 16- to 17-year-old secondary students’ CT skills in calculus lessons using the dynamic mathematics software GeoGebra. Our goal was to investigate how students experienced the CT-embedded calculus lessons with GeoGebra and what challenges they faced during their interaction with the learning material and software. We collected and analyzed data from students’ code in GeoGebra, workbooks, semi-structured interviews, and questionnaires. Our findings suggest that most students mastered using CT concepts in calculus activities to a satisfactory degree and could reason about their computational solutions using GeoGebra and the generated graphs. Students’ understanding of the mathematical content knowledge introduced was essential to complete the lesson series successfully and unnoticed gaps in prior knowledge emerged. Our study shows that students appreciate the CT-embedded calculus lessons and GeoGebra’s exploratory approach to mathematics problems when provided with appropriate support. We conclude that an integrated approach to mathematics education and CT is viable and can contribute not only to fostering CT but also to increasing interest in mathematics. PubDate: 2024-04-12

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Abstract: Abstract This article presents a case study that explores digital experiences in statistics teaching within Danish lower secondary school, focusing on the development of students’ statistical concepts. The study tracks the progress of a student named Frida, who engages with the digital tool TinkerPlots over the span of a year. Frida developed a unique ‘plot–stack–drag’ technique that significantly influenced her conceptual development during this period. Her routines with the tool not only supported her in some instances, but also created conflicts due to their impact on her personal goals and anticipations. This article delves into the educational implications of the dialectical relationship between students’ development of tool-based routines and their personal goals established during the process. The research findings highlight the profound impact of interactions between students and digital tools, such as TinkerPlots, on shaping students’ understanding of statistical concepts. This underscores the importance of educators’ heightened awareness of students’ personal goals and anticipations influenced by digital tools, which, in turn, opens the door to innovative learning opportunities. PubDate: 2024-04-05

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Abstract: Abstract The OЯTHO exhibit, situated within the LivingLab at Copernicus Science Centre (CSC) in Warsaw, Poland, introduces visitors to the Cartesian principles of representing numerical data as points in a two-dimensional space—the coordinate system. This interactive tabletop digital exhibit takes the form of a collaborative two-player game where participants manually control a virtual ball’s x- and y-axis values, respectively, to navigate it through maze-like paths, with the objective of completing the course in the shortest time and with the fewest moves. An action-based embodied design, OЯTHO draws on: (1) the enactivist tenet that individuals’ cognitive structures emerge from recurring task-effective sensorimotor patterns discovered through explorative perceptuomotor activity; and (2) cognitive-anthropological theorizations of shared ontologies as emerging through multimodal social interaction to facilitate the coordinated enactment of joint action. We overview the exhibit’s theoretical underpinnings and design conjectures; detail design challenges particular to the museum context, both in terms of the institution’s civic mandate and in terms of attracting and sustaining user engagement; present the exhibit itself; and speculate on the typology of research projects centered on mixed-methods multimodal data analysis based on expected 70,000 annual users. PubDate: 2024-04-03 DOI: 10.1007/s40751-024-00139-8

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Abstract: Abstract Can math concepts be experienced through the sensory modality of balance' Balance Board Math (BBM) is a set of pedagogical math activities designed to instantiate mathematical concepts through stimulation to the vestibular sense: an organ in the inner ear that detects our bodily balance and orientation. BBM establishes the different ways children spontaneously rock and move as the basis for inclusively exploring mathematical concepts together across diverse sensory profiles. I describe two activity sets where students explore focal concepts by shifting their balance on rockable balance boards: “the Balance Number Line,” using analog materials to foster understandings of the number line and negative numbers, and “Balance Graphing,” using sensors and a digital display to foster exploration of functions and graphing concepts, including the parameters of trigonometric functions and function addition. I outline proposed ways that engaging with concepts through balance-activating movement can change learners’ mathematical thinking and learning. PubDate: 2024-04-03 DOI: 10.1007/s40751-024-00140-1

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Abstract: Abstract In this article, the role of digital feedback that was provided in an outdoor mathematics education setting is taken into consideration. Using the app MathCityMap (2020) in the context of a mathematics trail, the influence of positive and/or negative feedback is examined in relation to how it influences the processes of verification and elaboration. In this context, special emphasis is placed on the students’ verification and elaboration and their relation to reasoning. In this qualitative study, 19 secondary students were filmed while solving mathematics tasks outdoors without digital support, as well as in indoor settings to enable a comparison. The results show that negative feedback in particular leads to a verification of the result. Still, an elaboration and explanation of why a result was incorrect was not often explicitly formulated by the students. Therefore, the potential of feedback is mainly seen in giving students a clear idea about the correctness of the result and searching for an alternative strategy to solve the task when in an outdoor setting. PubDate: 2024-03-04 DOI: 10.1007/s40751-024-00137-w

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Abstract: Abstract This study aimed to examine the quality of the open-access GeoGebra applets for function transformations by attending to the opportunity to learn function transformations afforded by them. Guided by the variation theory, this study conceptualized the opportunity to learn function transformations as the opportunity to discern critical features of function transformations. These critical features include function families in which function transformations are explored, representations of a parent function and its child function, and types and defining parameters of the transformations performed on the graph of a function. The results from the analysis show that the majority of existing GeoGebra applets use only one function family to explore a function transformation, of which quadratic and trigonometric functions are most prominent. The exploration of function translation is allowed in the largest number of applets followed by function dilation; function reflection over the x-axis, y-axis, and y = x; and function dilation. The functions in these applets are primarily represented symbolically and graphically and rarely have the defining parameters of transformations and corresponding points on the graphs of parent and child functions visible, which together suggest a graphical approach to function transformations. These results not only have implications for how to select existing GeoGebra applets for function transformations but also invite us to rethink how to design these applets that maximize students’ opportunity to learn function transformations conceptually in dynamic mathematical environments. PubDate: 2024-02-22 DOI: 10.1007/s40751-024-00136-x

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Abstract: This study aims to investigate students’ computational thinking (CT) through mathematical tasks integrated with programming in Scratch. Participants completed four tasks that required students to solve coding problems, which were focused on prime numbers and the prime factorization algorithm. The study was designed as a case study and the unit of analysis was the individual student. The study setting was a computer lab at a Science and Art Center, where students voluntarily enrolled after regular school hours. Data collected from three 7th- grade students included the screen- and audio-recordings of the environment and their screen, question-and-answer sessions and task sheets. Qualitative data were analysed through descriptive analysis based on Hoyles and Noss’s (2015) CT components: decomposition, abstraction, pattern recognition and algorithm design. Based on the findings, we observed that students tried to solve problems using the aforementioned components of CT. Through these programming tasks, we observed that, when they abstracted mathematical concepts, they designed the algorithm with different patterns. When the students could not make this abstraction, they carried out the algorithm design process by trial and error. We could claim that CT is related to the mathematisation encountered by the students during the abstraction process. PubDate: 2024-02-08 DOI: 10.1007/s40751-024-00135-y

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Abstract: Abstract The study advances the instrumental approach to mathematics education (Drijvers et al., 2013; Trouche, 2003), aiming to elucidate the interplay between students’ reasoning competency, conceptual knowledge and tool utilisation in dynamic digital geometry and algebra environments. The dynamic properties of these environments pose a nuanced predicament, as the outsourcing of translation between visual and algebraic representations raises concerns regarding students’ conceptual development and reasoning competency. To mitigate this issue, a prediction task is proposed, focusing on the dynamic behaviour of variable points in GeoGebra. I introduce a comprehensive framework adapting Toulmin’s argumentation model into the instrumental approach, emphasising processes of justification. This is complemented by the application of components of Vergnaud’s (1998) scheme concerning generative and epistemic ways to approach how students’ conceptual knowledge has played a part in these processes. Through a case study of a student pair solving a prediction task, I explore the links between instrumented justification, students’ mathematical reasoning competency and conceptual understanding, and how students’ use of GeoGebra tools is intertwined with their justification processes. The analysis reveals the intricate interplay between data production and interpretation, and it is grounded in inference drawn regarding students’ implied theorems about concepts, dynamic behaviour and progression in terms of techniques. The results indicate that the progression of technique is driven by the experience of the inefficiency of techniques and artefacts related to the goal of justification. Essentially, the framework links students’ reasoning competency to their use of tools and conceptual knowledge, as well as demonstrates that predicting dynamic behaviour can enhance knowledge-based justification. PubDate: 2024-01-29 DOI: 10.1007/s40751-024-00134-z

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Abstract: Abstract A premise of this article is that the current methods used in mathematics education research may be preventing researchers from adequately addressing the body and, in particular, the alignment of acting and knowing. Pursuing a non-dualistic and non-hierarchical approach to learning and knowing, I experiment with new methods that aim to increase situated and embodied validity. I do so through a short video clip of a four-year-old child interacting with TouchCounts, which is a multi-touch application designed to support early number sense. I work through the many arm, hand and finger actions made by the child, both manually on the screen and digitally in the air, focusing on the translations of these actions across contexts, which I understand as learning through remembering. I then discuss some consequences of these methods, which involve narrative and re-enactment, on knowledge production, both for learners and for researchers, specifically when digital technologies are involved. PubDate: 2023-11-14 DOI: 10.1007/s40751-023-00132-7

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Abstract: Abstract Many areas of mathematics naturally lend themselves to machine-based computing environments, which suggests that computational environments may serve as useful mediating tools for the teaching and learning of mathematical content. While some mathematics classes are leveraging the use of computational tools, the implementation of computer programming to teach and learn mathematics is not widespread. In this study, I highlight the mathematical activity of four undergraduate students who used Python to solve mathematical tasks in the context of set theory and logic. To understand how the students leveraged the computer programming environment, I use the analytical framework known as the instrumental approach, which can be utilized to investigate the confluence of an artifact (often a piece of technology) and the human mind to solve a mathematical problem. Results indicate that the students were able to use Python and its computational capabilities such as For Loops, If Statements, and Functions as artifacts to reason about propositional statements, set intersection, and subsets. Specifically, three instrumented action schemes emerged from their work on three different tasks. These schemes describe the use of Python in creative ways to solve mathematical tasks, which suggests various implications for teaching and research. PubDate: 2023-11-13 DOI: 10.1007/s40751-023-00130-9

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Abstract: Classification is a complex process that involves scientific concepts and higher-order mental processes such as abstraction, generalization, and pattern recognition. Even though it is an important competence for understanding the world, dealing with data and information, and solving complex problems, the education system embeds just its simplest operations and only in very early schooling. This study examines six middle school students’ activities as they play, modify, and redesign two Tetris-like classification games on the mathematical concepts of number sets and angle in an online authoring system called Sor.B.E.T (Sorting Based on Educational Technology). The qualitative data analysis of students’ dialogues aimed to bring in the foreground the classification processes students applied and the way these processes were entangled with the development of meanings and ideas on the mathematical concepts embedded in the games. According to the results, the play and modding of the two classification games enabled the development of higher-order classification processes such as objects’ properties comparison, properties discrimination, and classes’ encapsulation. They also supported meaning-making processes and triggered discussions about abstract mathematical notions, such as the concept of angle in various typical mathematical or physical contexts and the concept of number sets, the boundaries of each one, and the relationships among them through exploration and learner-generated exemplification. PubDate: 2023-10-25 DOI: 10.1007/s40751-023-00131-8

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Abstract: Abstract The slow uptake of technology by mathematics teachers is in contrast with the rapid growth in the availability of different digital resources specifically designed to help teaching and learning mathematics. We refer to platforms that were designed to permit for mathematical communication between multiple users. We seek to explore the affordances of such digital platforms to support mathematics teachers who wish to integrate technology as part of their practice, when planning and enacting technology-based mathematical activity. Specifically, we ask: What are the affordances and constraints of the platforms that may support instrumentation and instrumentalization processes leading to the development of teacher’s didactic instrument for planning and enacting a mathematical activity in a digital environment' The four platforms we chose for analysis are STEP, DESMOS, WIMS and Labomep. Our analysis shows on the one hand that the platforms afford support to the teacher while enacting technology-based mathematics activities. On the other hand, we suggest several components of didactic instrumental genesis that mathematics teachers need to develop in order to take benefit from digital platform affordances. These components include the ability to base decision-making on data gathered and visualised in dashboard embedded in learning management systems. PubDate: 2023-10-11 DOI: 10.1007/s40751-023-00127-4

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Abstract: Abstract Usable Math available at https://usablemath.org/ is being designed as an open online tutor for elementary school math students and teachers in the USA. It provides interactive problem solving for 3rd through 6th grade students who are learning mathematical reasoning and computation skills through word problems. Using the system, students and teachers access standardized test questions and then receive math learning strategies from four virtual coaches — Estella Explainer, Chef Math Bear, How-to Hound, and Visual Vicuna — who offer words, images, pictures, charts, graphs, animations, and gifs to support students to read, compute, and think strategically and visually as math problem solvers. This paper describes six key decisions influencing the design and development of the system: (1) four coaches, each with a different approach to math problem solving; (2) Google Slides as an online platform for the system; (3) a click-to-see method for students’ and teachers’ control of the system; (4) growth mindset feedback statements after every problem; (5) invitations for students to write their own math problems and problem solving strategies; and (6) ways for teachers to utilize the system for classroom learning. PubDate: 2023-10-06 DOI: 10.1007/s40751-023-00128-3

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Abstract: Abstract Rich research evidence supporting the strong, positive connections between spatial ability and mathematical ability has been well documented. Spatial tools, such as computer-aided-design (CAD) software, have been increasingly utilized and researched for promoting mathematics learning in K–12 (kindergarten to 12th grade). Tinkercad, a free online CAD program, offers a powerful spatial tool for expanding mathematics learning into space and improving mathematics learning through 3D modeling in an interactive virtual space. From the lens of embodied cognition research, the present paper illustrates the technological affordances of Tinkercad for leveraging the positive spatial-mathematical connections and discusses adopting action-based embodied design to create Tinkercad embodied spatial-mathematical learning experiences for elementary students. Specific Tinkercad embodied spatial-mathematical learning activities with closed-ended embodied actions and open-ended embodied actions are presented and implications for instruction and educational research are discussed. PubDate: 2023-09-29 DOI: 10.1007/s40751-023-00129-2

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Abstract: Abstract Computational thinking (CT) is gaining attention in education as a part of digital literacy and can be addressed in several disciplines, including mathematics. Through the lens of Brennan and Resnick’s framework, we investigated how computational concepts, practices, and perspectives can be addressed in upper-secondary statistics lessons using spreadsheets through design-based research. Three classes of, in total, 58 16- to 17-year-old 11th-grade students explored several authentic real-life data sets in three 2-h sessions using spreadsheets. We evaluated the intervention by analyzing students’ workbooks, spreadsheet files, interviews, and questionnaires. The findings indicate that (1) students successfully engaged in computational concepts through using formulas, parameters, and conditional statements, (2) fruitfully applied data practices, and (3) demonstrated awareness of the relevance of CT for their everyday and future lives. These results highlight the potential of the use of spreadsheets in secondary school for developing computational thinking skills. Implications for further integration of CT in the mathematics curriculum are discussed. PubDate: 2023-03-15 DOI: 10.1007/s40751-023-00126-5

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Abstract: Abstract We present a theoretical study that allows us to attempt framing in an embodied perspective the effectiveness of the drawing robot GGBot in the learning of geometry. The aim of the article is to set the intertwining of activity, semiotics, perception, and knowledge at the crossover of Radford’s theory of objectification (TO) and Borba and Villareal’s notion of humans-with-media. Such a crossover is articulated in four building blocks: (1) processes of objectification and the role of semiotic means of objectification, where we state that digital artifacts such as the GGBot change the topology of the semiotic means of objectification; (2) cognition is sensuous and learning is a process of domestication of the eye, where perception is theoretically shaped by the interaction with GGBot; (3) GGBot and humans-with-media, where we outline new thinking collectives and their modes of activity; (4) domestication of the eye triggered by transitions between domains of activity. Each building block of our theoretical discussion is empirically anchored to four episodes involving primary school students’ learning geometrical figures using the GGBot. To conclude, we focus on two basic concepts of geometrical thinking that unfold in the shift between domains. PubDate: 2023-02-15 DOI: 10.1007/s40751-023-00124-7

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Abstract: Abstract In this article, we draw on assemblage theory to investigate how children aged 5 engage with different material surfaces to explore ordinal and relational aspects of number. The children participate in an activity in which they first interact with a strip on the floor, then with a multi-touch iPad application, to work with numbers in expressive ways. Focus is on the physicality and materiality of the activity and the provisional ways that children, surfaces, and number come together. While the notion of assemblage helps us see how movement animates the mathematical activity, we enrich our understanding of the entanglement of children, matter, and number as sustained by coordinated movements, from which numbers emerge as relations. PubDate: 2022-12-13 DOI: 10.1007/s40751-022-00117-y