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?
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.
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.
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.
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?
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.