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Dynamic Mathematical Figures with Immersive Spatial Displays: The Case of Handwaver

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Abstract

We report on the design and development of HandWaver, a mathematical making environment for use with immersive, room-scale virtual reality. A beta version of HandWaver was developed at the IMRE Lab at the University of Maine and released in the spring of 2017. Our goal in developing HandWaver was to harness the modes of representation and interaction available in virtual environments and use them to create experiences where learners use their hands to make and modify mathematical objects. In what follows, we describe the sandbox construction environment, an experience within HandWaver where learners construct geometric figures using a series of gesture-based operators, such as stretching figures to bring them up into higher dimensions, or revolving figures around axes. We describe plans for research and future development.

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... In AR, learners also have direct access to three-dimensional figures rather than two-dimensional projections of three-dimensional figures. This could reduce cognitive load (Dimmel & Bock, 2019) and improve learning (Jacobson, 2011); this is an especially important affordance of XR (Johnson-Glenberg, 2018;Wu et al., 2013). Finally, in AR-based DGS, students can manipulate the scale of objects (Dimmel & Bock, 2019), making them very small to room-sized or bigger. ...
... This could reduce cognitive load (Dimmel & Bock, 2019) and improve learning (Jacobson, 2011); this is an especially important affordance of XR (Johnson-Glenberg, 2018;Wu et al., 2013). Finally, in AR-based DGS, students can manipulate the scale of objects (Dimmel & Bock, 2019), making them very small to room-sized or bigger. ...
... From this, we conclude that AR might be especially effective in getting an initial sense of the properties of 3D shapes This is consistent with past research that has used AR to help students learn about basic geometrical elements of 3D shapes (Andrea et al., 2019;Arvanitaki & Zaranis, 2020;Demitriadou et al., 2020;Gecu-Parmaksiz & Delialioglu, 2019). This also supports theories suggesting that being able to interact with shapes in three dimensions is an important affordance of XR (Dimmel & Bock, 2019;Jacobson, 2011;Johnson-Glenberg, 2018;Wu et al., 2013). There is less to be enthusiastic about when it comes to AR for reasoning about 2D shapes, based on our results. ...
Article
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Augmented Reality (AR) technologies allow for holograms to be layered over the real-world, “augmenting” human vision by adding technical information or illustrations onto 3D space. Although AR-based applications are showing positive effects in many systematic reviews and meta-analyses, well-designed, rigorous studies with strong control conditions are lacking. Further, many experimental studies lack process data to illuminate what is happening as students engage with AR. In this pre-registered study, we conducted an experiment where n = 120 high school students were assigned to reason about identical geometry simulations collaboratively either using tablets or AR head‐mounted displays (HMDs). We look at their learning and how it was impacted by the dimensionality (2D or 3D) of the shapes they explored, as well as how they engaged with virtual objects using gestures and epistemic actions. AR HMDs were more effective for students getting an initial sense of 3D shapes, but less effective for 2D shapes. For gaining insights into the workings of shapes and formulating justifications of conjectures, we see no evidence AR is more effective, and trends indicating AR may be detrimental to eliciting generalizations. Further, process data showed that students using tablets are more likely to manipulate the geometric shapes in the simulations, while students using the AR HMDs are more likely to use dynamic gestures that simulate these manipulations, which are less constrained by the objects’ actual properties. Implications for the future design and use of AR in education are given.
... AR can also allow students to easily explore geometric objects at many different scales (Dimmel & Bock, 2019). Thus you can have interactions that range from the scale of smaller handheld everyday objects (i.e., microspace-scale objects), to objects near the size of the human body like furniture or packing boxes that are at a room-scale (i.e., mesospace-scale objects), to very large objects like buildings, bridges, or natural environments (i.e., macrospace-scale objects; Bock & Dimmel, 2021a;Herbst & Boileau, 2018). ...
... In AR, learners have direct access to three-dimensional figures rather than two-dimensional projections of three-dimensional figures; this could reduce cognitive load (Dimmel & Bock, 2019) and improve learning (Jacobson, 2011). Johnson- Glenberg (2018) refers to this as one of the profound affordances of AR/VR (see also Wu et al., 2013). ...
... Further, it is important to critically consider the trade-offs of designing an environment where multiple students can see tethered objects in the same position, and both gesture and position their bodies with respect to the objects in mathematically meaningful ways for others to observe. Our study points to the importance of co-located mathematical objects to facilitate gestures in AR and VR environments, and the power of scale for being immersed in geometric objects (Dimmel & Bock, 2019). ...
... The purpose of this study was to fuse the spatiality of material explorations of worldly objects with the continuous variability of dynamic diagrams. To that end, we investigated digital representations of plane figures that were realized as spatial diagrams: diagrams that were inscribed in three-dimensional space, as opposed to on two-dimensional surfaces (Dimmel and Bock 2019). The spatial diagrams were rendered in an immersive environment that could be accessed by wearing a head-mounted (virtual-reality) display. ...
... In two-dimensional dynamic geometry software, we expect the perspective to have an orthogonal view of figures in a plane that may be scaled and translated; in three-dimensional dynamic geometry software, we expect that the view can be both translated and rotated throughout the space around the figure. Finally, recent developments in digital diagrams allow learners to vary their perspective by natural movement of their bodies (Google 2016;Neuhauser 2020;Dimmel and Bock 2019), taking on new perspectives in similar ways to learners engaging in walking scale geometry. ...
... Users could translate the vertices of the triangle along their respective lines by grasping them with their hands, whose locations in space were tracked using a Leap Motion sensor (see Figure 6; also, see Dimmel and Bock 2019, for details about the handtracking). Figure 6 shows the immersed user's view of a model of his hand, which is mapped to the physical position of his hand and used to interact with the dynamic figures. ...
Article
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In this article, we examine spatial inscriptions marked in real or rendered spaces, rather than on two-dimensional surfaces, conceptualize spatial inscriptions from an inclusive materialist perspective and consider realizations of spatial inscriptions that are possible with emerging technologies (e.g. 3D pens, immersive virtual reality). We then describe two cases of immersive environments that allowed learners to make and interact with spatial inscriptions. Next, we analyze how movements of participant–environment–inscription assemblages realized diagrams. Our analysis highlights how varying scale and changing perspective can become resources for doing mathematical work with spatial inscriptions.
... In the study detailed in this article, we explored two questions that concern spatial inscriptionsnamely those marked in real or rendered spaces, rather than on twodimensional surfaces (Dimmel and Bock 2019). How might spatially inscribed realizations of mathematical figures be encountered by learners? ...
... Gesture-tracking technologies make it possible to 'touch' (im)material spatial inscriptions, in the sense that they can be manipulated by making movements (e.g. pinching, grasping, pointing) with one's hands or body (Dimmel and Bock 2019). But the experiences of feeling mathematical structures described above are not yet possible. ...
... They are neither subject to laws that constrain the behavior of massive objects nor impeded by things in the world (Hart et al. 2017a, b;Kaufmann 2011).They combine the spatial presence of real things with the transformability of dynamic diagrams. They allow for representations of mathematical figures to be extended in three spatial dimensions (Kaufmann 2011), explored at various scales, and viewed from different perspectives (Dimmel and Bock 2019). ...
Article
Full-text available
In this article, we examine spatial inscriptions marked in real or rendered spaces, rather than on two-dimensional surfaces, conceptualize spatial inscriptions from an inclusive materialist perspective and consider realizations of spatial inscriptions that are possible with emerging technologies (e.g. 3D pens, immersive virtual reality). We then describe two cases of immersive environments that allowed learners to make and interact with spatial inscriptions. Next, we analyze how movements of participant–environment–inscription assemblages realized diagrams. Our analysis highlights how varying scale and changing perspective can become resources for doing mathematical work with spatial inscriptions.
... When learners manipulate mathematical objects (like triangles, lines, etc.) using gestures in VR environments, they may engage in various actions, including resizing, rotating, reflecting, constructing, and dilating. VR technologies allow learners to easily explore geometric objects at many different scales (Dimmel & Bock, 2019) -in VR, learners are able to make an icosahedron in the palm of their hand, or they could make one so large that they could walk inside of it. These manipulations all occur in a context where multiple learners can manipulate the same object at the same time and see each other's manipulations in real time. ...
... Collaborative body movements Body motions, defined as the way the learner moves their avatar (including the avatar's head) to engage in collaborative reasoning and problemsolving, are another key type of collaborative interaction in VR environments. For example, VR systems allow learners to change perspectives by moving their bodies, and as a result learners have direct access to 3D figures rather than 2D projections of 3D figures (Dimmel & Bock, 2019). Johnson-Glenburg (2018) refers to this as one of the two profound affordances of VR (see also Wu et al., 2013). ...
Article
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This study investigates how learners collaboratively construct embodied geometry knowledge in shared VR environments. Three groups of in-service teachers collaboratively explored six geometric conjectures with various virtual objects (geometric shapes) under the guidance of a facilitator. Although all the teachers were in different physical locations, they logged into a single virtual classroom with their respective groups and were able to see and manipulate the same geometric shapes as well as see their collaborators’ avatars and actions on the shapes in real time in the shared virtual space. This paper introduces a novel multimodal data analysis method for analyzing participants’ interactive patterns in collaborative forms of actions, gestures, movements, and speech. Results show that collaborative speech has a strong simultaneous relationship with actions on virtual objects and virtual hand gestures. They also showed that body movements and positions, which often focus on virtual objects and shifts in these movements away from or around the object, often signal key interactional collaborative events. In addition, this paper presents five emergent multimodality interaction themes showing participants’ collaborative patterns in different problem-solving stages and their different strategies in collaborative problem-solving. The results show that virtual objects can be effective media for collaborative knowledge building in shared VR environments, and that structured activity design and moderate realism may benefit shared VR learning environments in terms of equity, adaptability, and cost-effectiveness. We show how multimodal data analysis can be multi-dimensional, visualized, and conducted at both micro and macro levels.
... This capitalizes on embodied views of the nature of cognition (Lakoff & Núñez, 2000;Nathan, 2021), which posit that all conceptual knowledge is understood and experienced through the body and is action-based in nature. Embodied views of mathematical cognition often give rise to learning pedagogies where students engage in perceptual, sensorial, and motor activities to deepen their understanding of mathematical ideas (e.g., Abrahamson & Sánchez-García, 2016;Dimmel & Bock, 2019;Smith et al., 2014), including through gestures. ...
Conference Paper
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Virtual Reality (VR) provides unique learning experiences that allow students to immerse themselves in three-dimensional environments where they can explore mathematical ideas. Prisms VR is a suite of VR simulations for mathematics that allows students to have simulated real-world modeling experiences while working through topics such as ratios and proportionality. We examine affordances and constraints of VR for mathematics learning, focusing on manipulating and switching attention between different dynamic representations of mathematical principles, some of which involve immersion. Findings showed that some students enjoyed the immersive experiences and were able to be successful moving between the representational forms in VR, while others struggled with the overwhelming environment.
... The concrete nature of shAR objects may allow learners to better leverage prior knowledge and use new strategies (e.g., Goldstone & Son 2005). Students can also change their perspective by moving their body and are able to interact directly with 3D figures instead of 2D projections of 3D figures (Dimmel & Bock, 2019;Johnson-Glenburg, 2018). ...
... When gestural congruency is present, physical actions may create embodied resources and metaphors for students (Abrahamson & Sánchez-García, 2016;Smith, King, & Hoyte, 2014). AR also allows students to change perspectives by moving their bodies, and learners have direct access to three-dimensional figures rather than two-dimensional projections of three-dimensional figures; this could reduce cognitive load (Dimmel & Bock, 2019) and improve learning (Jacobson, 2011). Johnson-Glenberg (2018) refers to this as a profound affordance of AR. ...
Conference Paper
Full-text available
Novel forms of technology, like shared Augmented Reality (AR) holograms, can spur the discovery of new hypotheses about cognition and how it is embodied and distributed. These holograms have affordances for exploration, collaboration, and learning that have never been seen before as they were not previously possible. In the present study, we examine the multimodal ways that high school students interact with joint AR holograms while exploring geometric conjectures. Through multimodal analysis, we find both possibilities and important design considerations.
... Exemplarisch sei aus neuerer Zeit an dieser Stelle die Anwendung Handwaver von Bock und Dimmel (2020) genannt. Sie legten bei der Entwicklung der VR-Umgebung einen Fokus auf Handgestensteuerung und erforschen VR im Kontext von embodied cognition (Dimmel & Bock, 2019). Auf der Grundlage von Untersuchungen zu Gesten und ihrer Verbindung zu mathematischen Objekten implementierten sie in Handwaver verschiedene Möglichkeiten, mit Gesten mathematische Objekte zu konstruieren. ...
Chapter
Virtual Reality (VR), Augmented Reality (AR) und ihre Mischformen (MR/XR) sind durch die immer leistungsfähigeren Computer und mobilen Endgeräte (Tablets, Smartphones, mobile HMDs) inzwischen auch für den schulischen Einsatz verfügbar. In diesem Kapitel stellen wir Grundbegriffe, Technologien, Modelle und erste Umsetzungen von MR-Umgebungen im Mathematikunterricht vor.
... VR is increasingly used in the military (Champney, Stanney, Milham, Carroll, & Cohn, 2017), medicine (Li et al., 2017), tourism (Loureiro, Guerreiro, & Ali, 2020), and in educational settings for language learning (Kaplan-Rakowski & Wojdynski, 2018), mathematics (Dimmel & Bock, 2019), biology (Garcia-Bonete, Jensen, & Katona, 2019), and others. VR is also finding its way into the classroom at a practical level by offering experience needed in real-world scenarios, which can be more cost-effective than in-person and on-site training (Makransky & Lilleholt, 2018). ...
Chapter
This study investigated how the sense of presence and the plausibility illusion of high-immersion virtual reality (VR) impacted students' public speaking anxiety when presenting in a foreign language. In the study, the students gave eight presentations in a VR classroom while using a high-immersion VR headset. The students' virtual audience resembled classmates who were programmed to show nonverbal behavior, such as gestures, mimicry, and body motion. Analysis of subsequent individual semi-structured interviews with the students showed that they experienced a sense of presence and plausibility illusion about the virtual audience and the virtual space. The participants also saw VR as an effective tool for practicing public speaking and reducing any attendant anxiety.
... VR is increasingly used in the military (Champney, Stanney, Milham, Carroll, & Cohn, 2017), medicine (Li et al., 2017), tourism (Loureiro, Guerreiro, & Ali, 2020), and in educational settings for language learning (Kaplan-Rakowski & Wojdynski, 2018), mathematics (Dimmel & Bock, 2019), biology (Garcia-Bonete, Jensen, & Katona, 2019), and others. VR is also finding its way into the classroom at a practical level by offering experience needed in real-world scenarios, which can be more cost-effective than in-person and on-site training (Makransky & Lilleholt, 2018). ...
Chapter
Full-text available
This study investigated how the sense of presence and the plausibility illusion of high-immersion virtual reality (VR) impacted students' public speaking anxiety when presenting in a foreign language. In the study, the students gave eight presentations in a VR classroom while using a high-immersion VR headset. The students' virtual audience resembled classmates who were programmed to show nonverbal behavior, such as gestures, mimicry, and body motion. Analysis of subsequent individual semi-structured interviews with the students showed that they experienced a sense of presence and plausibility illusion about the virtual audience and the virtual space. The participants also saw VR as an effective tool for practicing public speaking and reducing any attendant anxiety.
... Dynamic geometry environments unite these affordances for plane figures; spatial diagrams offer a novel context for solid figures. In this study, I consider projected spatial inscriptions; projected spatial inscriptions are inscriptions in a three-dimensional space that "fill space but do not take up space" (Dimmel, Paniscio, Bock, submitted) and can be rendered with immersive spatial displays (Dimmel & Bock, 2019). Google's Tilt Brush is one example of a tool for constructing projected spatial inscriptions which "oppose the conventional encounter of a painting [inscribing] as a flat rectangular plane" (Chittenden, 2018, p. 389), where the representation of a three-dimensional figure is not distorted by projection or constrained to a specific point of view. ...
Article
Mithala and Balacheff (2019) describe three difficulties with two-dimensional representations of three-dimensional geometrical objects: “it is no longer possible to confuse the representation with the object itself,” visually observed relationships can be misleading, and analysis of the representation requires the use of lower-dimensional theoretical properties. Despite these difficulties, students are routinely expected to learn about three-dimensional figures through interacting with two-dimensional inscriptions. Three-dimensional alternatives include diagrams realized through various spatial inscriptions (e.g., Dimmel & Bock, 2019; Gecu-Parmaksiz & Delialioglu, 2019; Lai, McMahan, Kitagawa & Connolly, 2016; Ng and Sinclair, 2018). Such diagrams are three-dimensional in the sense that they occupy real (e.g., 3D pen drawings) or rendered (e.g., Virtual Reality/Augmented Reality environments) spaces as opposed to being inscribed or displayed on surfaces. Digital spatial diagrams can be grasped and transformed by gestures (e.g., stretching, pinching, spinning), even though they can’t be physically touched (Dimmel & Bock, 2019). Spatial diagrams make it possible to use natural movements of one’s head or body to explore figures from new perspectives (e.g., one can step inside a diagram), as they natively share the three-dimensional space. In this study I ask: How do learners use perspective to make arguments while exploring spatial diagrams? In particular, how do participants use perspectives outside and within geometric figures to make arguments while exploring spatial diagrams? To investigate this question, I designed a large-scale spatial diagram of a pyramid whose apex and base were confined to parallel planes. The diagram was rendered in an apparently unbounded spatial canvas that was accessible via a head-mounted display. The pyramid was roughly 1 meter in height and the parallel planes appeared to extend indefinitely when viewed from within the immersive environment. I created this diagram as a mathematical context for exploring shearing, a “continuous and temporal” measure-preserving transformation of plane and solid figures (Ng & Sinclair, 2015, p.85). I report on pairs of pre-service elementary teachers’ arguments about shearing of pyramids, using Pedemonte and Balacheff’s (2016) ck¢-enriched Toulmin model of argument. Shearing is a mathematical context that is likely novel to pre-service elementary teachers and provides an opportunity to connect transformations of plane and solid figures. Participants used perspectives outside and within the diagram to make arguments about the shearing of pyramids that would not be practicable with rigid three-dimensional models or dynamic two-dimensional representations. The results of this study suggest that the dimensionality of the spatial diagrams supported participants’ arguments about three-dimensional figures without mediation through projection or lower-dimensional components. The findings of this study offer a case that challenges the constraints of two-dimensional representations of three-dimensional figures, while maintaining theoretical constraints in a spatiographically accurate representation.
... Dynamic geometry environments unite these affordances for plane figures; spatial diagrams offer a novel context for solid figures. In this study, I consider projected spatial inscriptions; projected spatial inscriptions are inscriptions in a three-dimensional space that "fill space but do not take up space" (Dimmel, Paniscio, Bock, submitted) and can be rendered with immersive spatial displays (Dimmel & Bock, 2019). Google's Tilt Brush is one example of a tool for constructing projected spatial inscriptions which "oppose the conventional encounter of a painting [inscribing] as a flat rectangular plane" (Chittenden, 2018, p. 389), where the representation of a three-dimensional figure is not distorted by projection or constrained to a specific point of view. ...
Thesis
Full-text available
Mithala and Balacheff (2019) describe three difficulties with two-dimensional representations of three-dimensional geometrical objects: “it is no longer possible to confuse the representation with the object itself,” visually observed relationships can be misleading, and analysis of the representation requires the use of lower-dimensional theoretical properties. Despite these difficulties, students are routinely expected to learn about three-dimensional figures through interacting with two-dimensional inscriptions. Three-dimensional alternatives include diagrams realized through various spatial inscriptions (e.g., Dimmel & Bock, 2019; Gecu-Parmaksiz & Delialioglu, 2019; Lai, McMahan, Kitagawa & Connolly, 2016; Ng and Sinclair, 2018). Such diagrams are three-dimensional in the sense that they occupy real (e.g., 3D pen drawings) or rendered (e.g., Virtual Reality/Augmented Reality environments) spaces as opposed to being inscribed or displayed on surfaces. Digital spatial diagrams can be grasped and transformed by gestures (e.g., stretching, pinching, spinning), even though they can’t be physically touched (Dimmel & Bock, 2019). Spatial diagrams make it possible to use natural movements of one’s head or body to explore figures from new perspectives (e.g., one can stepinside a diagram), as they natively share the three-dimensional space. In this study I ask: How do learners use perspective to make arguments while exploring spatial diagrams? In particular, how do participants use perspectives outside and within geometric figures to make arguments while exploring spatial diagrams? To investigate this question, I designed a large-scale spatial diagram of a pyramid whose apex and base were confined to parallel planes. The diagram was rendered in an apparently unbounded spatial canvas that was accessible via a head-mounted display. The pyramid was roughly 1 meter in height and the parallel planes appeared to extend indefinitely when viewed from within the immersive environment. I created this diagram as a mathematical context for exploring shearing, a “continuous and temporal” measure-preserving transformation of plane and solid figures (Ng & Sinclair, 2015, p.85). I report on pairs of pre-service elementary teachers’ arguments about shearing of pyramids, using Pedemonte and Balacheff’s (2016) ck¢-enriched Toulmin model of argument. Shearing is a mathematical context that is likely novel to pre-service elementary teachers and provides an opportunity to connect transformations of plane and solid figures. Participants used perspectives outside and within the diagram to make arguments about the shearing of pyramids that would not be practicable with rigid three-dimensional models or dynamic two-dimensional representations. The results of this study suggest that the dimensionality of the spatial diagrams supported participants’ arguments about three-dimensional figures without mediation through projection or lower-dimensional components. The findings of this study offer a case that challenges the constraints of two-dimensional representations of three-dimensional figures, while maintaining theoretical constraints in a spatiographically accurate representation.
Conference Paper
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The emergence of immersive digital technologies, such as shared Augmented Reality (shAR), Virtual Reality (VR) and Motion Capture (MC) offers promising new opportunities to advance our understanding of human cognition and design innovative technology-enhanced learning experiences. Theoretical frameworks for embodied and extended cognition can guide novel ways in which learning in these environments can be understood and analyzed. This conceptual paper explores a research method in Educational Technology-multimodal analysis for embodied technologies-and provides examples from shAR, VR, and MC projects that use this approach. This analysis involves tracking learners' gestures, actions on physical and virtual objects, whole body movements and positions, and their talk moves, in addition to other relevant modalities (e.g., written inscriptions), over time and across space. We show how this analysis allows for new considerations to arise relating to the design of educational technology to promote collaboration, to more fully capture students' knowledge, and to understand and leverage the perspectives of learners.
Chapter
Virtual realityVirtual Reality (VR) provides an interesting environment to teach and learn 3D geometry. In this article, we discuss the use of Neotrie VRNeotrie VR as a 3D whiteboard for distance teachingDistance teaching that we have carried out during the 2020–21 academic year, with studentsStudent of the Mathematics degree at the University of Almería. We describe a concrete case on parametric equations ofParametric equations of surfaces surfaces, for which a 3D graphing calculator3D graphing calculator has been implemented, as well as a stereoscopic view camera to show 3D videos, which the studentsStudent can view with cheap stereoscopic glasses for mobile phones. From the side of the teacher, it is certainly much easier to explain 3D concepts on a 3D whiteboard like Neotrie than to use paper and pencil, blackboard, or any 2D digital tablet. StudentStudent feedbackFeedback is also analyzed after using various supports for manipulating and observing learning, includingGeoGebra GeoGebra, which can also serve to know how to use virtual realityVirtual Reality (VR) for distance learning.
Conference Paper
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One's affective state can change our actions and thought process while learning. The interplay of student's emotion and problem solving in spatial geometry has not been thoroughly studied. We present qualitative analysis of individual and collaborative problem solving of spatial geometry tasks by middle school students (8 students in a lab setting, and 21 students in group interviews in a classroom setting). We use the concept of embodied learning design, with the 3D printing pen as a medium, to make the process of converting 2D sketches to 3D models more explicit. Findings revealed that the students' affective state significantly influenced the way they solve the problems in spatial geometry. 3D sketching environment allows students to build a bond (intimacy) with the material and use their emotions as signals for heuristic changes (integrity). The discrepancy between 2D and 3D visualization in spatial geometry tasks may lead to students' emotional tension.
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We investigated preservice elementary teachers’ diagrammatic encounters with division by zero. Pairs of preservice teachers explored a transformable diagram where the locations of points on the x and y axes could be continuously varied. Quotients were defined in the diagram as the intersection of a line with the y-axis. For zero divisors, the quotient line was parallel to the y-axis, and there was no point of intersection. We report our analysis of two episodes where the transformability of the diagram spurred encounters with division by zero. In each episode, pairs of preservice teachers used repeated movements of the points in the diagram to explore the conditions under which the quotient line would become parallel to the y-axis. Our analysis shows how these movement-based material experiments gave rise to different conceptions of division by zero. We discuss how transformable diagrams create new material contexts for exploring arithmetic concepts.
Poster
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Objectives The state of the art for displaying and interacting with information is rapidly evolving. Technologies that allow users to interact with virtual, spatial figures have been commercially available since 2016 and are getting cheaper, smaller, and more feasible to implement in schools each month. By virtual, we refer to figures that are digitally rendered, such as a Geogebra diagram. By spatial, we refer to figures that exist in three-dimensional space, such as a chair in a room. Until recently, it was possible to explore fixed cases of spatial figures (e.g., a particular model of a platonic solid) or continuously transform two-dimensional virtual figures. But the commercial availability of what we call immersive spatial display technologies--e.g., virtual reality, augmented reality, holographic projection, and related technologies that render space itself as the canvass for making inscriptions (Dimmel & Bock, accepted for publication)--makes possible new types of mathematical representations: Dynamic, spatial figures that combine the extensionality of physical things with the malleability of virtual figures (see Figure 1). Research is needed to understand how these new technologies can affect the teaching and learning of mathematics. We report here a case study of how pre-service elementary teachers used dynamic, spatial figures to investigate two problems: (1) how does a triangle change as a vertex is moved on a line parallel to its opposite side? and (2) how does a pyramid change as it’s apex is moved on a plane parallel to its base? We studied the mathematical experiences of pre-service elementary teachers because the dynamic, three-dimensional figures that are possible with spatial displays could make concepts that are challenging for teachers more accessible. Tossavainen et al. (2016) suggest that PSETs understandings of the dimensionality of geometric measures are limited by investigations of plane figures with area formulas known to the PSETs (e.g., . A = ½ b*h, V = s^3). Tossavainen et al. (2016) recommend that PSETs broaden their investigations of measure to include “irregular and unbounded figures” (e.g. composite figures, regions bounded by curves). Shearing, an area-preserving transformation of some plane figures and volume-preserving transformation of some solid figures, is one opportunity for PSETs to explore limits of area of familiar geometric figures. Figure 1. Chloe and Danielle examining the sheared triangle after throwing its vertex. Figure 2. A triangle sheared to the boundaries of the screen in Geogebra Classic 6. With native control over perspective and a working space only limited by the participant’s gaze (Figure 1), immersive spatial displays allow for investigations of shearing to be extended to unbounded shearing that are difficult with traditional digital tools (Figure 2). We ask: What affordances of the designed environment do preservice elementary teachers use when investigating the effects of unbounded shearing of a triangle’s area and perimeter? Theoretical Framework Design-based research involves cycles of design and analysis to develop both learning environments and “prototheories of learning” in “authentic settings” (Design-Based Research Collective, 2003). We used the conjecture-mapping approach described by Sandoval (2014) to frame our study. Conjecture maps trace how high-level conjectures about learning can be embodied in designed environments whose mediating processes will produce desired outcomes. The specification of a conjecture map at the start of each design cycle is a means for the researcher to track how designs evolve from iteration to iteration (Sandoval, 2014). The high-level conjecture guiding our design is that new modes of interacting with virtual mathematical objects could help pre-service K-8 teachers broaden their conceptions of measure. This conjecture is grounded in our analysis of the affordances for natural, multimodal interaction that are germane to immersive, room-scale virtual reality environments (Dimmel & Bock, 2017). The case study reported below is the second-cycle of a design-based research project whose objective is to understand how an immersive virtual environment could support pre-service teachers’ reasoning about spatial figures. Below, we describe the virtual environment we developed in terms of key designed components that capitalized on the affordances of spatial displays (e.g., dynamic figures, immersive space, gesture interaction) In a prior study, participants used an earlier version of the environment (exclusive to shearing pyramids) and made claims about an analogous relationships between perimeter and area and surface area and volume (see: Bock & Dimmel, 2017). Based on this initial study, the revised version of the environment developed for this study was designed to provide virtual tools that could facilitate more in-depth investigations of how perimeter/area and surface area/volume are related. To describe participants’ mathematical activities as mediated by the designed immersive environment, we used the conceptions-knowing-concept (ckc) model (Balacheff & Gaudin, 2010; Vittori, 2018; DeJarnette, 2018). The cK¢ model provides a framework for describing a learner’s conceptions in terms of observable interactions within a mathematical learning environment or milieu. The model has four components: a general set of problems, a representation system, a set of operators, and a control structure. To assess the breadth of the PSETs conceptions, we can consider the general set of problems they were able to solve as an outcome. The representation system is generally constrained by the design of the virtual environment. The operators that participants use to act on the representation system (within the virtual environment) and the control structure they use to check their conclusions both function as mediating processes in the participants’ inquiry. Conjecture Map. The conjecture map highlights high level features of the design and their expected implications for mediating processes and outcomes. For this implementation, we consider an environment with size bounded by the user’s gaze, with the manipulatives described in the environment design. These manipulatives (described below) allow for the construction of non-prototypical triangles and pyramids: triangles that appear to approach line segments and pyramids that appear to approach polygons. The ability to construct such figures, immersive virtual workspace, and participant structure were designed to create conflict between participants about changes in the area and volume measure of the figures without numerical feedback that could be used in common formulas. Given extended time in the environment, we expected participants to find an internally satisfactory solution to the problems that they generated. Design of the Immersive Environment A virtual environment for use with immersive spatial displays was designed by the authors to allow pre-service teachers to explore measurement properties of triangles, pyramids, and prisms. We report here participants’ conceptions of shearing as mediated by two dynamic, spatial manipulatives. The first manipulative was a triangle with vertices bound to two parallel lines. The second manipulatives was a pyramid where the apex was bound to a plane parallel to the square base. Manipulative 1. An isosceles triangle was constructed on a plane orthogonal to the floor. A line was constructed through two vertices of the triangle (Figure 1). A second line was constructed parallel to the first passing through the remaining vertex. The two parallel lines were fixed, and the vertices were constrained to move along the parallel lines. Participants could use a pinch-and-drag operator to move any of the vertices along the line and a throwing operator to “send” the vertex at a constant speed along the line. Finally, a cross-section line segment would appear at the height of an open palm if placed adjacent to the triangle. In addition to the operators described above, there were two additional resources that could be introduced into the environment at the researcher’s discretion: a square whose area equaled the area of the triangle and a line segment whose length equaled the perimeter of the triangle. As the perimeter and area of the triangle changed, the area square and perimeter segment would change accordingly. The purpose of these spatial representations of measure was to provide a scaffold for comparing area and perimeter measures without numerical feedback that might encourage empirical conceptions of equality (Herbst, 2005). Once introduced into the scene, participants could move the spatial representations of measure by pinching them and placing them. Figure 3: A square-based pyramid with apex bound to a plane parallel to its base. Manipulative 2. A square based pyramid was constructed to have a base parallel to the floor (Figure 3). A plane was constructed parallel to the base of the pyramid through the apex of the pyramid. The base and plane were fixed, and the apex was constrained to move within the plane. Finally, a cube with volume equal to the pyramid and a square with area equal to the surface area of the pyramid can be enabled by the interviewers. Methods We investigated student conceptions of two- and three- dimensional shearing by conducting task-based, semi-structured interviews with pairs of pre-service K-8 teachers. Participants completed a background survey (demographic) and a short questionnaire that asked them questions about perimeter, area, surface area, and volume. After completing the survey and questionnaire, participants’ completed an orientation to immersive spatial displays. The orientation used Valve’s SteamVR Tutorial, which is designed to help participants get comfortable with the HTC Vive’s display technology and to introduce them to the basics of navigating a virtual space. The orientation also included a verbal description of the LeapMotion hand-tracking technology (see Materials) used to interact with the virtual environment. Second, participants worked in pairs during two 2-hour semi-structured interviews. During these interviews, one participant would be immersed in the environment while the other participant watched a live third-person mixed-reality composite image on a 70-inch display and a live first-person view from the immersed participant on a 22-inch display. Participants were able to switch roles or take off the head-mounted display at any time and were prompted to switch at least once during the investigation of each manipulative. Participants were asked to use a think-aloud protocol and collaboratively explore the figures. Interviewers prompted participants for additional detail about their reasoning but refrained from affirming or denying any mathematical claims that the participants made. Participant Selection. Participants were recruited from methods and mathematics content courses for pre-service elementary courses at a large university in New England and offered $15.00 compensation for each of three two-hour sessions that they attend. Fifteen PSETs completed the orientation session, eight completed the first interview and six students completed the second interview. Participants worked in pairs in each of the interviews, and were oriented in small groups. Materials. A desktop workstation equipped with two 1080p webcams, video capture cards, microphone and an HTC Vive with LeapMotion sensor. The LeapMotion sensor uses stereo infrared cameras to track hand positions, classify gestures, and render real-time virtual hands. Data Sources. During each of the interviews, audio recording captured the conversation between participants and the interviewers. Screen recordings captured a first person view of the environment from the immersed participant’s perspective and also a third person view of the immersed participant in real physical space. This real-world third person perspective was matched-to and blended with a third-person view of the virtual environment, which allowed us to capture a mixed-reality composite of the physical and virtual spaces (see Figure 1, above). Each of these recordings were synchronized and re-rendered into a master video recording, after audio was processed for noise reduction. Analysis Episodes where participants discussed unbounded shearing and area, perimeter, surface area or volume were selected for further analysis. These episodes were transcribed, participants’ names replaced with pseudonyms, and narrative summaries were written from the transcriptions describing the episode in the context of the interview. In this report, we discuss two of these selected episodes, with two different pairs of participants. Ashley and Brittany. Ashley and Brittany used their native control over perspective, local movement of the vertices of the triangle and the throwing operator to conclude "you can have the same area even if the perimeters are still wildly different" but were left with the question "if you have the same perimeter do you have the same area?” Ashley and Brittany solved the problem: Is there a one-to-one relationship between the perimeter and area of a triangle?. Chloe and Danielle. Initially, Chloe and Danielle predicted that the area of the sheared triangle will grow “once it goes past what it would be squished.” Chloe and Danielle used native control over perspective, the throwing operator, and visual inspection of graphical area measure to conclude that the as a triangle is sheared without bound, “it also gets thinner almost to the point where it's a line, or it looks like one … it is just stretching” using an analogy of rearranging grains of sand until the figure has a miniscule width. Chloe and Danielle solved the problem: Is a triangle’s area only conserved under shearing within a bounded region? Significance Both groups of pre-service teachers investigations illustrated conceptions supported by the immersive spatial displays. In both cases, the virtually unbounded three-dimensional space of the immersive spatial display allowed participants to investigate extreme cases of shearing. The native control of perspective allowed the second group to step into the plane of the 2-dimensional figure and watch the sheared triangle visually approach a line while the triangle’s base could simultaneously be observed with a finite length. These investigations were supported by affordances of immersive spatial displays that are difficult to replicate in traditional virtual or physical manipulatives. While immersive spatial displays make new representations of mathematical figures available to students, today’s PSETs (as tomorrow’s classroom teachers) need to have mathematically meaningful experience with immersive spatial displays in order to facilitate their future students’ own inquiry, as professional development and changes in beliefs alone may not be sufficient (Batane and Ngwako, 2017). This report begins to identify a set of affordances that could be leveraged in future design-based research studies to develop virtual environments to expand PSETs meaningful mathematical experiences.
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A room-scale virtual-reality environment was used to investigate students' conceptions of the volume of a pyramid. Participants controlled the virtual environment with a gesture-based interface that converted movements of their hands into actions on mathematical figures. Two students in graduate programs leading to certification in secondary science education investigated how the volume of a pyramid is affected by horizontal (i.e., shearing) or vertical (i.e., elongation) movements of its apex. Participants' actions within the environment were analyzed using the conceptions-knowing-concept (cK¢) model of student conceptions. Both participants used an analogy of volume to surface area and area to perimeter to make sense of the effects of the shearing operator.
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Tangible user interfaces (TUIs) are often compared to graphical user interfaces (GUIs). However, the existing literature is unable to demonstrate clear advantages for either interface, as empirical studies yielded different findings, sometimes even contradicting ones. The current study set out to conduct an in-depth analysis of the strengths and weaknesses of both interfaces, based on a comparison between similar TUI and GUI versions of a modeling and simulation system called “FlowBlocks”. Results showed most users preferred the TUI version over the GUI version. This is a surprising finding, considering both versions were equivalent in regard to most performance parameters, and the TUI version was even perceived as inferior to the GUI version in regard to usability. Interviews with users revealed this preference stemmed from high levels of stimulation and enjoyment, derived from three TUI properties: physical interaction, rich feedback, and high levels of realism. Potential underlying mechanisms for these findings and practical implications are discussed.
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Gestures are often taken as evidence that the body is involved in thinking and speaking about the ideas expressed in those gestures. In this article, we present evidence drawn from teachers' and learners' gestures to make the case that mathematical knowledge is embodied. We argue that mathematical cognition is embodied in 2 key senses: It is based in perception and action, and it is grounded in the physical environment. We present evidence for each of these claims drawn from the gestures that teachers and learners produce when they explain mathematical concepts and ideas. We argue that (a) pointing gestures reflect the grounding of cognition in the physical environment, (b) representational gestures manifest mental simulations of action and perception, and (c) some metaphoric gestures reflect body-based conceptual metaphors. Thus, gestures reveal that some aspects of mathematical thinking are embodied.
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This study explores interactions with diagrams that are involved in geometrical reasoning; more specifically, how students publicly make and justify conjectures through multimodal representations of diagrams. We describe how students interact with diagrams using both gestural and verbal modalities, and examine how such multimodal interactions with diagrams reveal their reasoning. We argue that when limited information is given in a diagram, students make use of gestural and verbal expressions to compensate for those limitations as they engage in making and proving conjectures. The constraints of a diagram, gestures and linguistic systems are semiotic resources that students may use to engage in geometrical reasoning.
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Formative assessment, in this article, is defined as "the process used by teachers and students to recognize and respond to student learning in order to enhance that learning, during the learning." The findings of a two-year research project in New Zealand indicate that formative assessment has the following characteristics: responsiveness, sources of evidence, a tacit process, using professional knowledge and experiences, an integral part of teaching and learning, formative assessment is done by both teachers and students, the purposes for formative assessment, the contextualized nature of the process, dilemmas, and student disclosure.