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Abstract: Abstract Using the theoretical constructs of double instrumental genesis and instrumental distance, in this article, I examine case studies of four primary school teachers (K–5) in British Columbia, Canada, who implemented the multi-touch, iPad application TouchTimes (hereafter, TT) into their mathematics teaching. This novel digital technology provides embodied and relational experiences of multiplication through two different dynamic multiplicative models. In interviews, these teachers shared their personal experiences learning about this relatively new digital application themselves, the obstacles they encountered and their experiences integrating TT into their instructional repertoires as a tool for student learning. My aim was to identify specific episodes in which transitions occurred during the implementation of technology-enhanced mathematics lessons and to highlight how instrumental distance influenced the teachers’ professional instrumental genesis. These episodes focus on (1) the internal shifts in thinking that the teachers experience personally and professionally while undergoing double instrumental genesis; (2) transitions across the two different microworlds that comprise TT, and the different multiplicative models portrayed by each of them; and (3) transitioning beyond the dynamic multiplicative models portrayed by TT towards mathematical activities with a static medium. My analysis indicates that these transitions are multi-faceted and complex, that the personal and professional instrumental geneses that teachers undergo may be closely intertwined and that, when speaking of TT, they clearly differentiate between ways of teaching with it and how students may learn using this technology. PubDate: 2023-04-01 DOI: 10.1007/s40751-022-00109-y
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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 Previously defined profiles for using a computer algebra system (CAS) describe prototypical uses of such a tool. When students use a dynamic geometry environment (DGE), however, new opportunities arise for transitions within and beyond the DGE. In this article, CAS profiles are used to explore students’ uses of a DGE. Four cases of students, all at the beginning of a task, are chosen to distinguish and discuss students’ various uses of a DGE, leading to four related profiles: ‘random’, ‘mechanical’, ‘rational’, and ‘theoretical–resourceful’. The students used a DGE template, which concerned functions as co-variation. The profiles include students’ various ways of dragging and their transitions within and beyond a DGE. The results indicate that difficulties with transitions within it and wandering dragging without testing conjectures are a characteristic of students showing a random profile. While students who show a mechanical profile also experience difficulties with transitions within the DGE, and they also are likely to refer to the DGE, which concerns the transitions beyond it. A rational profile characterises students who initially prefer paper and pencil work. Hence, a static understanding of functions may occur, and dragging may only appear for a short time. Finally, the theoretical–resourceful profile appears when students use the DGE for their purpose and do transitions within it and show different ways of dragging. PubDate: 2023-01-30 DOI: 10.1007/s40751-022-00123-0
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Abstract: Abstract This article focuses on the possible relationship between predictions, developed during the resolution of geometrical tasks within a Paper-and-Pencil Environment (PPE), and subsequent explorations within a Dynamic Geometry Environment (DGE). Building on Fischbein’s Theory of Figural Concepts, and to gain insight into the transition of predictions from the PPE to the DGE, I introduce an analytical scheme that includes the identification of the solvers’ figural concepts in focus and geometric predictions; dragging modalities that solvers make use of in the DGE; reactive and proactive dimensions of the interaction with dynamic objects in the DGE. I present a fine-grained qualitative analysis of data collected during task-based interviews as high school students reason about a geometrical task—first on paper-and-pencil and then in a DGE. The analyses of three representative examples show that the interaction with DGE objects can lead students to rethink the geometrical configuration and renegotiate predictions, depending on the circumstances that offer the opportunities to experiment with continuities or discontinuities during the transition of prediction from one to the other environment. PubDate: 2023-01-25 DOI: 10.1007/s40751-022-00119-w
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Abstract: Abstract This study focuses on a case study that highlights the mathematical discourse developed by two pairs of students when dealing with a specific transition task, i.e., an activity leading to the construction of a graph of a function based on the exploration of another representation of the same function. Such a task was designed to work on the “transition beyond” that involves moving from the graph of a function in a dynamic geometry environment to the Cartesian graph of the same function in the paper-and-pencil environment. In this case study, I analyze in fine-grained detail the discourse developed by two pairs of high-school students (ages 15–16) and describe how they translate the dynamism of the proposed representation into the paper-based context. The analysis aims at investigating the potentialities of transition tasks for supporting the building of bridges between multiple representations of the same function. The analysis also showcased the important role dragging routines played for making the transition. PubDate: 2023-01-03 DOI: 10.1007/s40751-022-00121-2
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Abstract: Abstract The introduction of digital artefacts for the teaching/learning of mathematics raises the issue of “transitioning” between the mathematical knowledge built in digital contexts and that built in a more traditional, non-digital context, and vice versa. In this article, we address this issue by presenting a teaching experiment aimed at fostering the development of the concept of variable in a lower secondary school, through the use of the spreadsheet. We depict a complex network of several intertwined transitions: from the non-digital to the digital context; between different activities within the same digital context; and from the digital to the non-digital context. We highlight the centrality of the teacher’s role in triggering and supporting these transitions, thus fostering the students’ construction of resources for solving the assigned tasks. For our analysis, we propose a conceptualisation of the resources that takes into account the fact that such resources cannot always be established a priori, but that they can emerge during the activity with the digital artefact or that they can even be constructed as an effect of it. Addressing this complexity allows us to deepen insights into how the classroom use of digital artefacts, the process of construction and emergence of resources, both for students and for the teacher, and different types of transitions nurture each other in a productive intertwining. PubDate: 2022-12-24 DOI: 10.1007/s40751-022-00122-1
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Abstract: Abstract Much important research on the learning of mathematics with technology-supported inquiry has been devoted to learning with multiple-linked representations (MLR) as a mode of feedback. Like a mirror, MLR feedback helps students see their actions in one representation “reflected” in another. Yet, research has followed learning episodes where MLR feedback did not lead to concept formation and the achievement of curricular goals. This article reports on the potential of what might be thought of as a mirror that speaks. In response to example-eliciting tasks, students use interactive diagrams to create examples to which mathematical descriptions are automatically associated. Such descriptions may be thought of as another kind of linked mathematical representation system. Transitions feature in two ways in our analysis of students’ use of this representation. At the level of student activity, we examine when students move between attending to textual descriptions and to the graphs that they describe. We are also interested in how attention to these descriptions and co-ordination with their own use of these words can support students in making a transition in their thinking from considering distance as only total distance traveled, to a co-ordinated view of distance including both total distance traveled and distance from a starting point. This article focuses on two example-eliciting motion tasks and two sets of descriptive words. We found that these sets of words helped students, while and after they were working with the diagram, to distinguish between total distance traveled and position with respect to a starting point. PubDate: 2022-12-20 DOI: 10.1007/s40751-022-00120-3
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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
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Abstract: Abstract Given the prevalence and academic impact of technology in mathematics education, increasing attention has been paid to the role of digital artifacts in promoting productive classroom discourse. This study investigates the association between the qualities of mathematics discussion and artifacts generated by individual teachers, individual students, and jointly by multiple students. We also examine the influence of pedagogical moves on artifact mediation and discourse. Drawing on video data from the mathematics classroom of five pre-service teachers, we found that discourse based upon student ideas occurred most frequently with reference to student-generated artifacts, and dialogic discourse occurred almost exclusively with reference to jointly constructed artifacts. With six episodes mediated by three types of artifacts, we provide descriptive empirical evidence to illustrate the complex interplay of artifact construction, student ownership, teacher pedagogical moves, and artifact mediation for mathematical discourse and meaning-making. We also provide an Artifact Mediated Discourse Matrix with examples as a framework for examining artifact-supported discourse in mathematics classrooms. We discuss implications related to teacher pedagogical moves, a classroom culture discourse, and the role of mediation embedded in the digital artifact. PubDate: 2022-12-01 DOI: 10.1007/s40751-022-00114-1
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Abstract: We investigate three 8th-grade students’ mathematical meanings developed in the context of using linked representations to generate animations of figural models tuned in musical rhythm in “MaLT2,” a programmable Turtle Geometry in 3D resource affording dynamic manipulation of variable values. We adopted a modified version of the UDGS (Using, Discriminating, Generalizing, Synthesizing) model, introduced by Hoyles and Noss in 1987, in order to frame and analyze students’ mathematical meaning-making process involving setting out goals; posing conjectures; using mathematical ideas to test them; and exploring, generalizing, and expanding these ideas. This dynamic process was contextualized and connected to a flow of two different types of transitions: (1) transitions within the different representations of MaLT2 and (2) transitions beyond MaLT2, among the representational contexts of the digital microworld, artistic ideas, and abstract mathematics. In our analysis, we use this theoretical concept to trace the kind of mathematical meanings connected to multidisciplinary notions embedded in dance and music, such as synchronicity, symmetry, periodicity, and harmony, emerging from this learning context. We also look into the way these mathematical meanings were gradually evolved from being implicitly integrated in digital and artistic ideas to being reflected on and generalized. PubDate: 2022-11-23 DOI: 10.1007/s40751-022-00118-x
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Abstract: Abstract Learning-support system is an umbrella term that we use for digital resources that assign students with mathematical questions and give automatic feedback on the inserted answers. Transitioning between questions and feedback is characteristic to students’ work with such systems. We apply the commognitive framework to explore the role of within-system transitions in students’ mathematics learning, with a special interest in what we term as “reroutinization”—a process of repeated development of conventional routines to be implemented in already familiar mathematical tasks. The study revolves around a digital module in integral calculus, which was designed to support undergraduates with finding areas enclosed by functions. The data comes from dyads and triads of first-year university students, who collaboratively interacted with the module. The analyses cast light on how transitioning within the module aided students to review familiar routines, amend them, confirm, and solidify the amendments. The transition process was not always linear and contained instances of students cycling back and forth between the assigned questions and feedback messages. We conclude with the discussion on the module’s design that afforded reroutinization and suggest paths for further research. PubDate: 2022-11-23 DOI: 10.1007/s40751-022-00116-z
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Abstract: Abstract We address a problem of promoting instructional transformation in early undergraduate mathematics courses, via an intervention incorporating novel digital resources (“techtivities”), in conjunction with a faculty learning community (FLC). The techtivities can serve as boundary objects, in order to bridge different communities to which instructors belong. Appealing to Etienne Wenger’s Communities of Practice theory, we theorise a role of the instructor as a broker, facilitating “boundary transitions” within, across, and beyond a set of digital resources. By “boundary transition”, we mean a transition that is also a brokering move; instructors connect different communities as they draw links between items in their instruction. To ground our argument, we provide empirical evidence from an instructor, Rachel, whose boundary transitions served three functions: (1) to position the techtivities as something that count in the classroom community and connect to topics valued by the broader mathematics community; (2) to communicate to students that their reasoning matters more than whether they provide a correct answer, a practice promoted in the FLC; (3) to connect students’ responses to mathematical ideas discussed in the FLC, in which graphs represent a relationship between variables. Instructors’ boundary transitions can serve to legitimise novel digital resources within an existing course and thereby challenge the status quo in courses where skills and procedures may take precedence over reasoning and sense-making. PubDate: 2022-11-09 DOI: 10.1007/s40751-022-00113-2
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Abstract: This article investigates the knowledge arising in mathematics teachers’ planning of how to manage transitions within and beyond dynamic geometry environments in the topic of circle theorems. The notion of situated abstraction is used to elaborate the central TPACK construct within mathematics education and address previous criticisms of the framework, specifically to clarify the distinction between the central construct and the dyadic constructs. Four case-study teachers each participated in a semi-structured interview based upon a pre-configured GeoGebra file. The teachers were asked to demonstrate how they would use the GeoGebra file to introduce students to the circle theorem that the angle at the centre of the circle, subtended by an arc, is double the angle at the circumference subtended by the same arc. The visual and audio aspects of the GeoGebra interviews were recorded and the TPACK framework used to analyse teachers’ knowledge arising in the four interviews. The central TPACK construct is illustrated with examples of teachers’ strategies for capitalising on transitions within and beyond dynamic geometry environments for the purposes of teaching circle theorems and contrasted with the dyadic construct of TCK. The utility of the theoretical elaboration of the TPACK construct within mathematics education is demonstrated and implications discussed. PubDate: 2022-11-07 DOI: 10.1007/s40751-022-00115-0
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Abstract: Abstract Incorporating visual representations, rather than strictly relying on symbolic representations, is a research-based strategy for supporting fraction learning. However, students must also make transitions between visual and symbolic fraction representations to apply the conceptual understanding they gained from visual representations to symbolic fraction computation. Virtual manipulatives (VMs) offer opportunities for supporting students in making these transitions, as many fraction VMs integrate visual and symbolic representations into one manipulative. Some VMs also dynamically link the representations, so learners can observe how changes to one representation impact the other. For these features to support students in transitioning among representations, teachers must orchestrate opportunities for students to use and reflect on their use of the features. This study examined how six fourth- and fifth-grade teachers orchestrated opportunities in lesson plans for students to make transitions among the visual and symbolic representations within and beyond fraction VMs. Results showed that teachers used two strategies for orchestrating these transitions: VM choice and direct teacher intervention. Implications of teachers’ uses of these strategies are discussed in terms of what kinds of transition opportunities were made available to students and what professional learning experiences could be needed to support teachers in orchestrating transitions. PubDate: 2022-09-27 DOI: 10.1007/s40751-022-00111-4
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Abstract: Abstract Instructional designs that include two or more artifacts (digital manipulatives, tables, graphs) have shown to support students’ development of reasoning about covarying quantities. However, research often neglects how this development occurs from the student point of view during the interactions with these artifacts. An analysis from this lens could significantly justify claims about what designs really support students’ covariational reasoning. Our study makes this contribution by examining the “messiness” of students’ transitions as they interact with various artifacts that represent the same covariational situation. We present data from a design experiment with a pair of sixth-grade students who engaged with the set of artifacts we designed (simulation, table, and graph) to explore quantities that covary. An instrumental genesis perspective is followed to analyze students’ transitions from one artifact to the next. We utilize the distinction between static and emergent shape thinking to make inferences about their reorganizations of reasoning as they (re-)form a system of instruments that integrates previously developed instruments. Our findings provide an insight into the nature of the synergy of artifacts that offers a constructive space for students to shape and reorganize their meanings about covarying quantities. Specifically, we propose different subcategories of complementarities and antagonisms between artifacts that have the potential to make this synergy productive. PubDate: 2022-09-20 DOI: 10.1007/s40751-022-00112-3
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Abstract: Abstract Research suggests that tasks that use technology as a reorganizer (technology is used to produce dynamic representations) are linked with the development of students’ conceptual knowledge, yet many secondary mathematics textbooks in the USA predominantly include tasks that use technology as an amplifier (technology is used to produce static images). Thus, if teachers wish to incorporate tasks that use technology as a reorganizer, they must locate these resources elsewhere or construct them themselves. This study took place in a technology, pedagogy, content course where each prospective teacher (PT) engaged in at least one component of an ideation, rehearsal/refinement, enactment, and reflection intervention, where they were asked to adapt traditional textbook lessons to promote students’ rich conceptual understandings of function transformations via technology as reorganizer tasks. A total of 15 PTs agreed to participate, 8 seeking a grade K-8 certification and 7 seeking a grade 6–12 certification. On the initial lesson plan involving linear function transformations, the majority of PTs (13 out of 15) used technology as an amplifier, but on the final lesson plan involving absolute value function transformations, 13 PTs used technology as a reorganizer. There were also increases in the incidence of lesson elements that promoted moderate or rich forms of conceptual knowledge between the initial and final lesson plans. We discuss the implications of these results and introduce readers to dynamic conceptual components as manipulable bridges between different mathematical representations that hold the potential to develop students’ richer forms of conceptual knowledge. PubDate: 2022-08-18 DOI: 10.1007/s40751-022-00110-5
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Abstract: Abstract There has been little written about the ways in which teachers understand draggable geometric objects in a dynamic geometry environment with respect to variance and invariance—two important mathematical ideas related to the development of spatial perception and geometric reasoning. In this article, we investigated such understanding from two perspectives: example spaces and continuous variation of an object. We found that teachers tend to generate many examples as related to invariance; however, they did not tend to use maintaining dragging or to enact continuous movements, as well as only few descriptions included links to generalization. We found that continuous variation is characterized by descriptions of an object: having invariant properties under continuous movement; having different instantiations/formations; containing strong links to generalization. We also found that teachers’ descriptions of draggable geometric objects were mostly associated with generating examples. This finding raises concerns about teachers’ understanding of draggable geometric objects as being mainly associated with one perspective, generating examples, that is more static in nature than dynamic. Possible implications for both teacher education and future research are discussed. PubDate: 2022-07-06 DOI: 10.1007/s40751-022-00106-1
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