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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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... 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. ...

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). ...

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.

... 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). ...

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). ...

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. ...

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. ...

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.

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.

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.

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.

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.

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.

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.

Whereas emerging technologies, such as touchscreen tablets, are bringing sensorimotor interaction back into mathematics learning activities, existing educational theory is not geared to inform or analyze such learning. In particular, educational researchers investigating instructional interactions still need intellectual and methodological frameworks for conceptualizing, designing, facilitating, and analyzing how students’ immersive hands-on dynamical experiences become formulated within semiotic registers typical of mathematical discourse. We present case studies of tutor–student behaviors in a technologically enabled embodied-interaction learning environment, the Mathematical Imagery Trainer for Proportion. Drawing on ecological dynamics—a blend of dynamical-systems theory and ecological psychology—we examine how students develop effective sensorimotor schemes as solutions to interaction problems. We argue that in attempting to enact a new complex skill, learners may improve their performance by spontaneously constructing an attentional anchor—a real or imagined object, area, or other aspect or behavior of the perceptual manifold that facilitates motor-action coordination. We further argue that symbolic artifacts introduced into the arena may both mediate new affordances for students’ motor-action control and shift their discourse into explicit mathematical re-visualization of the environment. We thus implicate symbolic artifacts as ontological hybrids evolving from things you act with to things you think with. Students engaged in embodied-interaction learning activities are first attracted to symbolic artifacts as prehensible environmental features optimizing their grip on the world, yet in the course of enacting the improved control routines, the artifacts become frames of reference for establishing and articulating quantitative systems known as mathematical reasoning.

The ability to mentally rotate objects in space has been singled out by cognitive scientists as a central metric of spatial reasoning (see Jansen, Schmelter, Quaiser-Pohl, Neuburger, & Heil, 2013; Shepard & Metzler, 1971 for example). However, this is a particularly undeveloped area of current mathematics curricula, especially in North America. In this article we discuss what we mean by mental rotation, why it is important, and how it can be developed with young children in classrooms. We feature results from one team of teacher-researchers in Canada engaged in Lesson Study to develop enhanced theoretical understandings as well as practical applications in a geometry program that incorporates 2D and 3D mental rotations. Children in the Lesson Study classrooms (ages 4–8 years) demonstrated large gains in their mental rotation skills during 4 months of Lesson Study intervention in the Math for Young Children research program. The results of this study suggest that young children from a wide range of ability levels can engage in, and benefit from, classroom-based mental rotation activities. The study contributes to bridging a gap between cognitive science and mathematics education literature.

This article and my career as an educational researcher are grounded in two fundamental assumptions: (a) that research and practice can and should live in productive synergy, with each enhancing the other, and (b) that research focused on teaching and learning in a particular discipline can, if carefully framed, yield insights that have implications across a broad spectrum of disciplines. This article begins by describing in brief two bodies of work that exemplify these two fundamental assumptions. I then elaborate on a third example, the development of a new set of tools for understanding and supporting powerful mathematics classroom instruction—and by extension, powerful instruction across a wide range of disciplines.

Three studies were conducted with middle school students to evaluate a web-based intelligent tutoring system (ITS) for arithmetic and fractions. The studies involved pre and post test comparisons, as well as group comparisons to assess the impact of the ITS on students' math problem solving. Results indicated that students improved from pre to post test after working with the ITS, whereas students who simply repeated the tests showed no improvement. Students who had more sessions with the ITS improved more than those with less access to the software. Improvement was greatest for students with the weakest initial math skills, who were also most likely to use the multimedia help resources for learning that were integrated into the software.

This paper outlines the new opportunities that that will be changing the landscape of geometry education at the primary school level. These include: the research on spatial reasoning and its connection to school mathematics in general and school geometry in particular; the function of drawing in the construction of geometric meaning; the role of digital technologies; the importance of transformational geometry in the curriculum (including symmetry as well as the isometries); and, the possibility of extending primary school geometry from its typical emphasis on vocabulary (naming and sorting shapes by properties) to working on the composing/decomposing, classifying, comparing and mentally manipulating both two- and three-dimensional figures. We discuss these opportunities in the context of historical developments in the nature and relevance of school geometry. The aim is to motivate and connect the set of papers in this special issue.

The primary goal of the study was to explore first-grade children’s reasoning about plane and solid shapes across various kinds of geometric representations. Children were individually interviewed while completing a shape-matching task developed for this study. This task required children to compose and decompose geometric figures to identify geometric shapes that either matched or did not match the stimulus shape. The stimulus shapes were 2D diagrams of plane and solid-shape geometric figures. The results showed that children overestimated the significance of triangular vertices (“pointiness”); certain kinds of scaling demands gave children trouble in shape classification; children had trouble translating lines found in 2D diagrams into 3D visual boundaries, especially where projected curvature was involved; and that children had difficulty reasoning consistently across the task. Implications for future research as well as teaching recommendations are discussed.

Intelligent Tutoring Systems (ITS) are computer programs that model learners’ psychological states to provide individualized instruction. They have been developed for diverse subject areas (e.g., algebra, medicine, law, reading) to help learners acquire domain-specific, cognitive and metacognitive knowledge. A meta-analysis was conducted on research that compared the outcomes from students learning from ITS to those learning from non-ITS learning environments. The meta-analysis examined how effect sizes varied with type of ITS, type of comparison treatment received by learners, type of learning outcome, whether knowledge to be learned was procedural or declarative, and other factors. After a search of major bibliographic databases, 107 effect sizes involving 14,321 participants were extracted and analyzed. The use of ITS was associated with greater achievement in comparison with teacher-led, large-group instruction (g = .42), non-ITS computer-based instruction (g = .57), and textbooks or workbooks (g = .35). There was no significant difference between learning from ITS and learning from individualized human tutoring (g = –.11) or small-group instruction (g = .05). Significant, positive mean effect sizes were found regardless of whether the ITS was used as the principal means of instruction, a supplement to teacher-led instruction, an integral component of teacher-led instruction, or an aid to homework. Significant, positive effect sizes were found at all levels of education, in almost all subject domains evaluated, and whether or not the ITS provided feedback or modeled student misconceptions. The claim that ITS are relatively effective tools for learning is consistent with our analysis of potential publication bias.

This article describes the initial implementation of an innovative program for elementary-age children involving origami and pop-up paper engineering to promote visuospatial thinking. While spatial ability measures correlate with science, technology, engineering, and math (STEM) success, a focus on spatial thinking is all but missing in elementary school education. Fourth-grade students took part in the program and then completed spatial thinking assessments or completed the assessments prior to program participation. All students completed assessments at three points in time—before, during, and after the intervention. Results suggest the program's promise in promoting spatial thinking, showing both spatial thinking gains and extensive engagement in the program. Questionnaire responses suggest the program may have particular appeal for girls, which could play a role in reducing gender disparities in spatial reasoning and in situations where spatial thinking can be applied.

Science educators have been interested in developing people's understanding of the epis-temology of science for a long time. Despite decades of research on students' ideas, and decades of instructional reforms, it remains very hard to change students' ideas about the nature of scientific knowledge and practice. There are essentially two specializations in science education related to epistemology: research on student's conceptions of the nature of professional science (NOS) and research on students' own efforts to make scientific meaning of the world through inquiry, modeling, argumentation, and so on. They have pro-duced different conclusions about students' understanding of the epistemology of science (Sandoval, 2005), and the fact that they are in very little dialogue with each other is a major obstacle to producing a coherent theory of epistemological development; a point I expand on below. Without such a theory, however, it is difficult to see how instructional experiences in school are likely to help students develop productive understandings of the epistemology of science. There are, at least, two reasons why we should want all students to leave high school with productive understandings of the epistemology of science. One is so that people understand what makes science science, to distinguish science as a field of human endeavor from other human endeavors. The second reason is to be able to use this understanding to identify scientific arguments and explanations from other kinds, and to be able to evaluate them in The ideas here have developed through work supported by National Science Foundation award #0733233, although they are my own and do not reflect the official views of the agency. C 2014 Wiley Periodicals, Inc.

In this paper, we use the work of philosopher Gilles Châtelet to rethink the gesture/diagram relationship and to explore the ways mathematical agency is constituted through it. We argue for a fundamental philosophical shift to better conceptualize the relationship between gesture and diagram, and suggest that such an approach might open up new ways of conceptualizing the very idea of mathematical embodiment. We draw on contemporary attempts to rethink embodiment, such as Rotman’s work on a “material semiotics,” Radford’s work on “sensuous cognition”, and Roth’s work on “material phenomenology”. After discussing this work and its intersections with that of Châtelet, we discuss data collected from a research experiment as a way to demonstrate the viability of this new theoretical framework.

It is well-documented that most students do not have adequate proficiencies in inquiry and metacognition, particularly at deeper levels of comprehension that require explanatory reason- ing. The proficiencies are not routinely provided by teachers and normal tutors so it is worth- while to turn to computer-based learning environments. This article describes some of our re- cent computer systems that were designed to facilitate explanation-centered learning through strategies of inquiry and metacognition while students learn science and technology content. Point&Query augments hypertext, hypermedia, and other learning environments with ques- tion-answer facilities that are under the learner control. AutoTutor and iSTART use animated conversational agents to scaffold strategies of inquiry, metacognition, and explanation con- struction. AutoTutor coaches students in generating answers to questions that require explana- tions (e.g., why, what-if, how) by holding a mixed-initiative dialogue in natural language. iSTART models and coaches students in constructing self-explanations and in applying other metacomprehension strategies while reading text. These systems have shown promising results in tests of learning gains and learning strategies.

In this note we formulate a didactic principle that can govern the use of symbolic computation software systems in math courses. The principle states that, in the treatment of each subarea of mathematics, one must distinguish between a "white-box" and a "black box" phase. In the "white box" phase, algorithms must be studied thoroughly, i.e. the underlying theory must be treated completely and algorithmic examples must be studied in all details. In the black box phase, problem instances from the area can be solved by using symbolic computation software systems. This principle can be applied recursively.

Although teachers today recognize the importance of integrating technology into their curricula, efforts are often limited
by both external (first-order) and internal (second-order) barriers. Traditionally, technology training, for both preservice
and inservice teachers, has focused on helping teachers overcome first-order barriers (e.g., acquiring technical skills needed
to operate a computer). More recently, training programs have incorporated pedagogical models of technology use as one means
of addressing second-order barriers. However, little discussion has occurred that clarifies the relationship between these
different types of barriers or that delineates effective strategies for addressing different barriers. If pre- and inservice
teachers are to become effective users of technology, they will need practical strategies for dealing with the different types
of barriers they will face. In this paper, I discuss the relationship between first- and second-order barriers and then describe
specific strategies for circumventing, overcoming, and eliminating the changing barriers teachers face as they work to achieve
technology integration.

The purpose of this study was to examine the direct and indirect effects of teachers’ individual characteristics and perceptions
of environmental factors that influence their technology integration in the classroom. A research-based path model was developed
to explain causal relationships between these factors and was tested based on data gathered from 1,382 Tennessee public school
teachers. The results provided significant evidence that the developed model is useful in explaining factors affecting technology
integration and the relationships between the factors.
KeywordsTechnology integration-Computer use-Technology use-Computer use in education-Path model

Recognizing geometric shapes and noting their special properties are important steps in learning geometry. Much of what you want students to learn about squares, rectangles, and trapezoids can be accomplished with geoboard figures. Some of what is lost in generality is gained in accuracy of construction, whether with pegs and rubber bands or with pencil and dot paper. If segments should be parallel, perpendicular, equal—or whatever—they look that way on geoboards.

Geometry class can be a lot of fun. An animated discussion with the students was unfolding.

Beliefs referring to teaching and learning mathematics with technology play an important role when teachers are integrating technology into their classrooms. However, there has been a lack of instruments to measure those beliefs in detail. In this paper, we contribute a detailed inventory to measure technology-related beliefs of in-service and pre-service teachers. This instrument—a questionnaire—is analyzed with data from 246 pre-service and 199 in-service teachers using confirmatory factor analysis. It is found that beliefs of in-service teachers can be measured in more detail than those of pre-service teachers, mainly due to a longer experience and a correspondingly more differentiated system of beliefs.

IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching) is an exciting new series of texts for teacher education which aims to advance the learning and teaching of mathematics by integrating mathematics content with the broader research and theoretical base of mathematics education. The Learning and Teaching of Geometry in Secondary Schools reviews past and present research on the teaching and learning of geometry in secondary schools and proposes an approach for design research on secondary geometry instruction. Areas covered include: teaching and learning secondary geometry through history; the representations of geometric figures; students' cognition in geometry; teacher knowledge, practice and, beliefs; teaching strategies, instructional improvement, and classroom interventions; research designs and problems for secondary geometry. Drawing on a team of international authors, this new text will be essential reading for experienced teachers of mathematics, graduate students, curriculum developers, researchers, and all those interested in exploring students' study of geometry in secondary schools. © 2017 Patricio Herbst, Taro Fujita, Stefan Halverscheid, and Michael Weiss. All rights reserved.

We present a flexible algorithmic pipeline for high-quality three-dimensional acquisition of dynamic real world objects. In this context we discuss the reduction of mesh complexity as one of the key challenges for visualizing reconstructed three-dimensional content in augmented and virtual reality applications.

We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulations of three-dimensional hyperbolic space are available at http://h3.hypernom.com.

What kinds of curriculum materials do mathematics teachers select and use, and how? This question is complex, in a period of deep evolutions of teaching resources, with the proficiency of online resources in particular. How do teachers learn from these materials, and in which ways do they ‘tailor’ them for their use and pupil learning? Teachers collect resources, select, transform, share, implement, and revise them. Drawing from the French term « ingénierie documentaire », we call these processes « documentation ». The literal English translation is « to work with documents », but the meaning it carries is richer. Documentation refers to the complex and interactive ways that teachers work with resources; in-class and out-of-class, individually, but also collectively.

This study investigated the ways in which the technological tool, The Geometer's Sketchpad, mediated the understandings that high school Honors Geometry students developed about geometric transformations by focusing on their uses of technological affordances and the ways in which they interpreted technological results in terms of figure and drawing. The researcher identified different purposes for which students used dragging and different purposes for which students used measures. These purposes appeared to be influenced by students' mathematical understandings that were reflected in how they reasoned about the physical representations, the types of abstractions they made, and the reactive or proactive strategies employed.

Geometry diagrams use the visual features of specific drawn objects to convey meaning about generic mathematical entities. We examine the semiotic structure of these visual features in two parts. One, we conduct a semiotic inquiry to conceptualize geometry diagrams as mathematical texts that comprise choices from different semiotic systems. Two, we use the semiotic catalog that results from this inquiry to analyze 2,300 diagrams from 22 high school geometry textbooks in which the dates of publication span the 20th century. In the first part of the article, we identify axes along which the features of geometry diagrams can vary, and in the second part of the article, we show the viability of using the semiotic framework to conduct empirical studies of diagrams in geometry textbooks.

The character of early mathematics education has changed over the last century not only in terms of the pedagogical approaches to teaching young children, but also in relation to the content and goals of that instruction. In this chapter, we explore psychological, sociocultural, and neurophysiological developments that may have helped to shape these pedagogical trends in early mathematics education. For example, early childhood educators have traditionally advocated for learning through play, but views of its nature and purpose have varied markedly among educational researchers and practitioners. In part, these variations exist as a function of the changing beliefs about children’s cognitive capabilities and about the role of early educational experiences in enhancing the capabilities of young minds. They also mirror shifting psychological orientations toward learning and philosophical perspectives toward knowledge and knowing. An examination of the writings of such influential theorists as Dewey, Thorndike, Piaget, Vygotsky, Flavell, and Rogoff serve to illuminate these shifting orientations toward young minds, mathematics, and the learning of mathematics. Drawing on these writings, we chart the course of early mathematics education in relation to these theoretical underpinnings, consider emerging trends, and address the implications for early mathematics research and practice.

There is a long-standing and ongoing debate about the relations between conceptual and procedural knowledge (i.e., knowledge of concepts and procedures). Although there is broad consensus that conceptual knowledge supports procedural knowledge, there is controversy over whether procedural knowledge supports conceptual knowledge and how instruction on the two types of knowledge should be sequenced. A review of the empirical evidence for mathematics learning indicates that procedural knowledge supports conceptual knowledge, as well as vice versa, and thus that the relations between the two types of knowledge are bidirectional. However, alternative orderings of instruction on concepts and procedures have rarely been compared, with limited empirical support for one ordering of instruction over another. We consider possible reasons for why mathematics education researchers often believe that a conceptual-to-procedural ordering of instruction is optimal and why so little research has evaluated this claim. Future empirical research on the effectiveness of different ways to sequence instruction on concepts and procedures is greatly needed.

This article traces some of the influential ideas and motivations that have shaped a large part of the research on the use of new technologies in mathematics education over the past 40 years. Particular attention is focused on Papert's legacy, Celia'sHoyles' transformation of it, and how both relate to the current research landscape that features not only dramatic changes in hardware and software, but also new philosophical perspectives on the embodied nature of the learner and ofmathematics. © The Author 2014. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved.

To situate the contributions of these research articles on visualization as an epistemological learning tool, we have employed mathematical, cognitive and functional criteria. Mathematical criteria refer to mathematical content, or more precisely the areas to which they belong: whole numbers (numeracy), algebra, calculus and geometry. They lead us to characterize the “tools” of visualization according to the number of dimensions of the diagrams used in experiments. From a cognitive point of view, visualization should not be confused with a visualization “tool,” which is often called “diagram” and is in fact a semiotic production. To understand how visualization springs from any diagram, we must resort to the notion of figural unity. It results methodologically in the two following criteria and questions: (1) In a given diagram, what are the figural units recognized by the students? (2) What are the mathematically relevant figural units that pupils should recognize? The analysis of difficulties of visualization in mathematical learning and the value of “tools” of visualization depend on the gap between the observations for these two questions. Visualization meets functions that can be quite different from both a cognitive and epistemological point of view. It can fulfill a help function by materializing mathematical relations or transformations in pictures or movements. This function is essential in the early numerical activities in which case the used diagrams are specifically iconic representations. Visualization can also fulfill a heuristic function for solving problems in which case the used diagrams such as graphs and geometrical figures are intrinsically mathematical and are used for the modeling of real problems. Most of the papers in this special issue concern the tools of visualization for whole numbers, their properties, and calculation algorithms. They show the semiotic complexity of classical diagrams assumed as obvious to students. In teaching experiments or case studies they explore new ways to introduce them and make use by students. But they lie within frameworks of a conceptual construction of numbers and meaning of calculation algorithms, which lead to underestimating the importance of the cognitive process specific to mathematical activity. The other papers concern the tools of mathematical visualization at higher levels of teaching. They are based on very simple tasks that develop the ability to see 3D objects by touch of 2D objects or use visual data to reason. They remain short of the crucial problem of graphs and geometrical figures as tools of visualization, or they go beyond that with their presupposition of students' ability to coordinate them with another register of semiotic representation, verbal or algebraic.

This paper challenges the ‘romantic’ view of the evolution of writing and the role of the Greeks in the development of alphabetic literacy. It introduces the concept of ‘autoglottic space’ and argues that this affords a better explanation of how writing restructures thought. At the same time it emphasizes that such questions cannot be decontextualized from the social and political circumstances attendant upon the introduction of writing into particular cultures, nor from the diverse purposes which writing may serve.

During the first 12 weeks of an applied calculus course, two classes of college students ( n =39) studied calculus concepts using graphical and symbol-manipulation computer programs to perform routine manipulations. Only the last 3 weeks were spent on skill development. Class transcripts, student interviews, field notes, and test results were analyzed for patterns of understanding. Students showed better understanding of course concepts and performed almost as well on a final exam of routine skills as a class of 100 students who had practiced the skills for the entire 15 weeks.

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.

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.

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.

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.