1 Introduction

Unprecedented investments are being made in tutoring interventions for K-12 students, predominantly because of issues stemming from the COVID-19 pandemic (e.g., the American Rescue Plan; The White House, 2022), and recent advances in generative AI that can be applied to tutoring (e.g., OpenAI 2024). Here we report a study of an afterschool Virtual reality (VR) mathematics tutoring intervention for grades 7–8 students, with the goal of learning how to leverage VR environments to support students’ mathematical reasoning. This study takes place in the context of research and practice on math tutoring, embodied learning, and virtual reality.

A mathematics tutoring model that is rapidly being developed and scaled is for students to engage in speech and/or text-based tutoring from a distance with a human tutor (e.g., Sass and Ali 2023; Zhang et al. 2023) or an AI tutor (e.g., Khan Academy’s “Khanmigo” or ASSISTments’ “CAIT”). Distance tutoring is often desired due to a lack of personnel and resources located physically at school and the efficiency with which it can operate. Recent results from human online mathematics tutoring are not always promising (Sass and Ali 2023; Schueler and Rodriguez-Segura, 2023; Zhang et al. 2023; Zydney and Hord 2023). Early results from generative AI math tutoring are also not always promising (Cheng et al. 2024).

One important issue with online tutoring is that it often treats learning as disembodied (Nathan 2023; Shvarts and van Helden 2023) and does not account for physically co-located bodies as important to the learning process. Theories of embodied cognition have arisen in education, arguing that all knowledge, even knowledge in domains traditionally considered abstract and hierarchical like mathematics, is rooted in bodily systems like perception, movement, gesture, and simulated action (Lakoff and Núñez 2000). VR is one way to bring opportunities for embodiment into online tutoring that has received increased attention recently due to its potential for distance learning in the wake of the COVID-19 pandemic (Mado et al. 2022).

VR is a 3D multimedia environment where learners interact with a computer-generated world (Onyesolu and Eze 2011)—this world is often fully immersive (Makransky and Petersen 2021). Technologies defined as VR vary widely and can involve handhelds, computer screens, projection, motion capture, or head-mounted displays (HMD) like goggles. A profound affordance of VR is the opportunities it provides for students to physically embody concepts and principles through mechanisms like gestural congruency and perception of immersion (Johnson-Glenberg 2018). Many VR HMDs have advanced hand-tracking that allows students to accurately see each other’s motions and gestures in real time, along with head and body position. Meta-analyses of VR in education show positive effects of VR, with immersive forms of VR most effective (Villena-Taranilla et al. 2022).

While VR could be a promising direction for distance tutoring, there are few K-12 studies of VR that use HMDs, have educators in the VR environment, involve mathematics tutoring, or involve extended sessions (Hamilton et al. 2021; Pellas et al. 2020). At the intersection of these needs is an opportunity to explore educator-facilitated, HMD-based VR for math tutoring. The rise of affordable and portable VR equipment allows remote tutors, located anywhere in the world, to interact with students in the embodied, first-person manner that is normally reserved for face-to-face tutoring, over extended periods of time. VR tutoring also has affordances that face-to-face tutoring does not—like being immersed with dynamic mathematical objects, where students can use a touch interface to manipulate those objects in ways impossible in the real world (Dimmel et al. 2021; Hohenwater and Fuchs 2004).

We studied an online learning environment where small groups of students would wear VR headsets to join an adult mathematics tutor in a virtual classroom where users were displayed as avatars. This classroom contained dynamic geometric figures that could be interacted with and manipulated through touch in the virtual environment. Our research questions examine (1) the effects of VR-based math tutoring on middle school student math achievement, compared to a control group that does not receive math tutoring, and (2) the novel affordances and constraints that VR math tutoring has, based on an analysis of video footage from the tutoring sessions.

2 Theoretical framework and literature review

2.1 Embodied cognition and gesture

Theories of embodied cognition posit that all knowledge is perceptual and action-based in nature (Lakoff and Núñez 2000). Wilson (2002) describes embodied cognition as a stance rooted in cognitive science that accentuates sensory and motor functions in human interaction with the world. Precisely defining embodied cognition is difficult, as there is a diversity of claims about the nature of embodiment made by different theorists. However, Wilson (2002) outlines six claims of embodied cognition theories that have been most prominent, and that can guide an integrated definition of embodied cognition. These claims include that cognition is situated actively in real-world environments, cognition is enacted within the environment in real time, cognition can be offloaded onto the environment by embodied interactions like actions and gestures, that the environment is part of the cognitive system over which cognition is distributed, that cognitive mechanisms serve as a way to support situation-appropriate actions, and that all cognition, even cognition seemingly decoupled from the environment, is still situated in past experiences interacting with the world.

When considering embodied cognition in education, Nathan (2021) proposes that embodied learning involves connecting unfamiliar ideas to learners’ lived experiences, perceptions, and body-based actions, and that such connections are needed for educational experiences to be meaningful. Interacting with instructional materials using movements can promote generative learning where students make connections and offload cognitive demands, which are key mechanisms through which embodied approaches can enhance learning (Castro-Alonso et al. 2024). Little research has examined embodiment in a distance education context; one exception is Shvarts and van Helden (2023), who found some students could successfully leverage embodied principles obtained during independent practice, while others needed the guidance of a tutor.

Gestures, spontaneous or purposeful movements of the hands that learners use (Hostetter and Alibali 2019), can support reasoning. Pointing gestures, where students use their hands to indicate objects or positions, as well as iconic gestures, where students represent objects with their hands, may be particularly important (Alibali and Nathan 2012). In a tutoring context, learners and tutors may point to objects (e.g., point to the vertex on a diagram of a triangle) or represent mathematical objects (e.g., make a triangle using their thumbs and forefingers) as they solve problems. Interlocutors can also use collaborative gestures where they explicitly build on the gestural reasoning of interactional partners (Walkington et al. 2019)—for example, one learner may echo (copy) another learner’s gesture of a 90° angle. Research suggests that such collaborative gestures can both enhance communication and allow for new ideas to emerge in collaborative groups (Walkington et al. 2019). Learners can also be tasked with creating their own gestures to correspond to mathematical ideas and then disseminate these gestures to create shared meaning around mathematical principles. For example, a tutor might ask students to create a hand gesture with their virtual hands that shows what “parallel” means.

In VR environments, an important consideration is whether learners experience a “sense of embodiment,” defined as “the ensemble of sensations that arise in conjunction with being inside, having, and controlling a body” (Kilteni et al. 2012, p. 374–375). In VR environments, sense of embodiment includes the feeling that one’s body is located inside of one’s VR avatar in terms of its spatial location and the feeling that one has agency in controlling one’s movements and predicting their consequences. Agency over one’s actions and movements may be a particularly important affordance of VR (MicGivney, 2025). Sense of embodiment also includes a person’s sense of body ownership which can involve considerations like morphological similarity between the avatar body and their real body. Avatar representations that do not resemble real bodies may be useful in some mathematical situations, however (e.g., having 4 fingers when learning base-8 counting; Chatain et al 2023). Body ownership in VR has been found to be negatively correlated with individuals’ body awareness (Chatain et al. 2022), and individuals with high body awareness (like dancers) may struggle more to position themselves in VR environments (Chatain et al. 2024).

2.2 Mathematics tutoring

One-on-one tutoring in a traditional (face-to-face) context has been shown to be effective for student learning. Bloom (1984) reviewed the literature and determined that one-on-one tutoring had an effect size of d = 2.0, which he referred to as the “2 sigma problem” (p. 6), because of the difficulty of getting such an effect using other more efficient approaches. However, Nickow et al. (2020) in a recent meta-analysis of face-to-face tutoring interventions, found an effect size of d = 0.37, with slightly stronger effects for professional tutors and during-school programs, but no benefit for tutoring being one-on-one rather than higher student-tutor ratios. Pellegrini et al. (2021) conducted a meta-analysis of 22 tutoring programs in elementary mathematics, of which only 1 was online. The online tutoring program had the lowest effect size of − 0.03, while one-on-one tutoring from teachers had an effect size of 0.22.

Pellegrini et al.’s (2021) result echoes other recent studies of online tutoring, which have become common since the COVID-19 pandemic. Zhang et al. (2023) examined a system for online tutoring of grade 7 mathematics and found that usage was low and impacts were small. They also found that the tutoring was not effective for learners with lower prior knowledge. Zydney and Hord (2023) describe an online tutoring program where pre-service teachers tutored high school students with learning disabilities in mathematics. They found that document cameras which allowed tutors to see students’ gestures atop their paper, were helpful, but the lack of document cameras allowing students to see tutors’ gestures was a major limitation. Schueler and Rodriguez-Segura (2023) found no positive effects of a grades 3–6 phone-based math tutoring intervention, with some evidence of detrimental effects. Sass and Ali (2023) found null effects for the use of the tutor.com online program. However, Gortzar et al. (2024) implemented an online mathematics tutoring intervention in grades 7–8 which significantly improved both standardized test scores and student grades. Positive effects on grades only occurred for students who had a good internet connection. Overall, the literature is mixed, with ample indication that online mathematics tutoring may not be effective.

2.3 VR in mathematics education

One way in which mathematics tutoring may be made more effective is through the utilization of VR. A number of studies have explored more generally the use of VR to engage students in mathematics learning. Su et al. (2022) engaged high school students in three VR simulations for learning geometry, finding increases in learning motivation and learning outcomes. Çakıroğlu et al. (2023) engaged middle school students in VR mathematics activities and found important increases in mathematical literacy through student discussion and knowledge co-construction. Akman and Çakır (2023) examined fourth grade students learning about fractions in VR, and did not find overall differences in performance or engagement, but did find positive effects of VR for affective engagement specifically. When considering the comparison of Augmented Reality to VR, Demitriadou et al. (2020) found that elementary students learning about geometric solids through either AR or VR learned significantly more than those using paper materials. However, AR and VR were not significantly different from each other for learning outcomes. Dimmel et al. (2020) explored an immersive, open environment for allowing adults to explore spatial and geometric figures finding evidence that VR can allow learners to better visualize spatial and geometric concepts, and that changing the scale of figures in VR offers new opportunities for mathematical understanding.

A meta-analysis of VR studies in K-6 mathematics found 5 studies and reported an overall effect size of d = 0.52 (Villena-Taranilla et al. 2022). A scoping review of VR in STEM education (Pellas et al. 2020) concluded that there was a lack of longitudinal studies of VR, a lack of mathematics studies, and that many VR interventions lacked a way for the instructor to be present to give support. In addition, very few studies reported on the challenges of using VR, and HMDs were almost never used in K-12 settings. Our study addresses many of these gaps.

In addition to omitting teachers, few VR environments allow for student–student collaboration. Embodied instructional approaches leverage social cognition, where performing movements and gestures and watching the movements and gestures of others can aid communication and learning (Castro-Alonso et al. 2024). Virtual objects in VR can enhance communication as learners do not have to rely on imagination to illustrate dynamic objects (Pidel and Ackermann 2020). In addition, learners have direct access to three-dimensional figures rather than 2D projections of 3D figures (Dimmel and Bock 2019). These figures can be explored at different scales, including scales so large that learners can be immersed inside mathematical objects (Dimmel and Bock 2019).

Whole body movements where students walk around mathematical objects (Walkington et al. 2024a) or navigate through a Cartesian coordinate plane, can also be important embodied resources in VR. For example, a student may walk around a pyramid and look at it from above and below to gain different perspectives. Figures can be explored from different perspectives that are impossible when considering typical diagrams; this can “foster a correspondence between mathematical concepts and how we move our bodies in space” (Dimmel et al. 2020, p. 16). Research suggests that users are able to spatially orient themselves similarly well in the real world versus in VR (Pastel et al. 2022). Research also suggests that learners with higher spatial ability scores tend to experience less disorientation in VR and can better use spatial strategies in VR, suggesting that spatial ability enhances the VR experience (Gittinger and Wiesche, 2024).

VR also allows opportunities for embodiment and play. Like gameplay, the social/recreational play and rapport building are beneficial to student engagement with the interface (Farrow and Iacovides 2014; Kateros et al. 2015). Particularly, active manipulation of, and control over physical and virtual objects (e.g. realistic hands or avatars synced with real body movements) is known to strengthen the integration of those objects and the action-space with one’s body schema (Maravita and Iriki 2004; Chandrasekharan 2009). It is likely that the recreational games some of our tutors incorporated in their instruction helped strengthen/fine-tune the students’ embodied relationships with their avatars (Roth and Latoschik 2020), thus, also potentially contributing to their overall sense of immersion and presence (Farrow and Iacovides 2014). Further, embodiment, immersion, and presence have been positively linked to better overall engagement and learning (e.g. Maresky et al., 2019). Therefore, the incorporation of social/recreational play is a direct design implication for VR.

While there are many promising elements of VR, there are also downsides to this technology. Parong and Mayer (2018) found that an immersive VR biology lesson was more distracting than a typical slideshow, resulting in higher extraneous cognitive load and emotional arousal, and lower learning outcomes. Johnson-Glenberg et al. (2021) found that VR may be detrimental to learning if learners are not given agency and control of their environment. Some research suggests that VR environments should mainly be used for learning about 3D objects, and that studying 2D objects in these spaces may be detrimental or lead to confusion (Walkington et al. 2024b). The way in which VR environments are designed is key to determining their effectiveness.

2.4 Dynamic geometry software

Some VR programs for mathematics use dynamic geometry software (DGS). DGS allows students to manipulate and interact with mathematical objects like shapes and figures (Hollebrands 2007). Students can use “dragging” motions (where they click on or touch a position on a figure) to change the position of points on a figure and see how relationships and measurements are updated in real time (Leung et al. 2013). Some implementations of DGS can allow for collaborative co-action where learners position themselves around mathematical objects (Hegedus and Otálora 2023). When DGS is touch-based, the haptic dragging actions can allow students to understand invariant mathematical properties (Pittalis and Drijvers 2023) in an embodied way. Importing DGS into VR environments is a natural extension of this work, given that VR environments are ideal for engaging in dynamic manipulations.

DGS can allow learners to engage in mathematical play, which involves agentic learners taking mathematical actions and observing the environment’s reactions, experiencing failure in the environment, understanding causality of one’s actions and the environment’s reactions, and then anticipating the environment’s reactions and connecting them to prior learning (Williams-Pierce and Thevenow-Harrison 2021). DGS can facilitate play because they “offer various modes of feedback that can enable students to experiment, test conjectures, and debug without having to appeal to an outside authority” (Williams-Pierce et al. 2019, p. 1979). A VR-based DGS may be ideal to facilitate mathematical play, because of the novel interactions students can have in the environment that are impossible in the real world (Walkington et al. 2024a).

2.5 Research purpose and questions

Research on online mathematics tutoring in the wake of the COVID-19 pandemic has not been promising. However, leveraging embodied learning approaches in a VR environment may allow for a new vision of distance math tutoring that involves action, gesture, and mathematical play. Here we report a research study that was conducted within the paradigm of embodied design: “embodied-design investigations seek both to evaluate the purchase of embodiment theory in educational research and, reciprocally, to utilize the iterative, cyclic method of design practice—ideate, build, implement, evaluate, re-theorize, and over again—as an empirical context for conducting studies poised to elaborate on embodiment theory” (Abrahamson et al. 2021, p. 3). We explore using VR HMDs to support middle school math learning in a 7-week tutoring study.

We report both opportunities and significant challenges of VR-based mathematics tutoring. Our research questions are:

  1. (1)

    What are the effects of VR-based math tutoring on middle school student math achievement, compared to a control group that does not receive math tutoring?

  2. (2)

    What novel (a) affordances, and (b) constraints, does VR math tutoring have?

Our study addresses several important gaps in the literature, in order to confront the problem of how to make mathematics tutoring more effective and embodied when presented from a distance. We report on the application of VR to mathematics, which has been understudied, focusing on the area of geometric and spatial reasoning where the 3D nature of VR has the potential to be uniquely leveraged. We examine HMDs which offer important opportunities for gesture-based reasoning through their immersiveness and hands-free interface. We have a human tutor present with the students in the VR environment, and students co-present with each other, allowing for a better understanding of person-to-person interaction and the potential of VR for collaborative forms of embodiment. In addition, our study examines students’ use of VR over time, allowing the potential to see the development of mathematical reasoning and interaction; this is an important element when considering VR in the context of tutoring, and many past VR studies look only at very short duration interventions (Hamilton et al. 2021; Pellas et al. 2020; for an exception, see, e.g., McGivney, 2025). We also explicitly look at the challenges of implementing VR-based interventions, as unlike many other studies we collected rich qualitative data of student and tutor interactions related to learning the content.

3 Methods

3.1 Participants and procedure

Participants included n = 38 students in grade 7 (24 students) and grade 8 (14 students), of whom 20 were randomly assigned to the experimental group receiving VR math tutoring, and 18 were assigned to the control group. Eighteen students identified as female and 20 as male. Participants were recruited from after-school programs at two urban schools where almost all students qualified for free/reduced lunch. Twenty-seven students identified as Hispanic, 10 as Black, and 1 as White. Students’ self-reported grades in math class varied widely (10 As, 12 Bs, 12 Cs, and 4 Ds), suggesting that these were typical students in their grade levels.

For students in the experimental group, seven 30–60 min tutoring sessions were held after school (Appendix 1), with the average student attending 54% of the sessions. Students met in the gym at their schools, where Oculus Quest 2 VR HMDs were available. They joined virtual classrooms where a tutor physically located at a local university would be virtually present; this tutor was usually an undergraduate student. Students would typically be grouped with 1–2 other students and would choose avatars to appear as. These avatars had hands that would display the students’ gestures through advanced hand-tracking, and heads and torsos that would show how the student’s body was positioned. Students could also target and teleport to different locations in the VR environment using specific hand movements.

The tutoring environment was designed by GeoGebra (Hohenwater and Fuchs 2004), a company that distributes a free, widely-used DGS. Tutors could open different “simulations,” which would display dynamic mathematical objects that were shared (everyone was in the environment together and could work with the same objects). The 7th grade activities covered prisms and pyramids, while the 8th grade activities covered reflections, rotations, translations, and dilations (Table 1). Students could manipulate the mathematical objects using their hands, with or without a room-sized Cartesian coordinate grid displayed.

Table 1 Images and descriptions of different VR simulations used during tutoring
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The control group also did VR tutoring activities in the Oculus Quest headsets, taking identical pre- and post- math assessments, but the VR tutoring activities they received related to English Language-Arts (ELA) content instead of mathematics content. The control group activities were delivered by the same organizational team (but involved different tutors), were video-recorded in the same manner, and occurred for the same total amount of time on the same days. Students in the control group had the same requirements for parental consent. A different choice for a control group would have been to have a control group of students who received mathematics tutoring outside of VR, and compare VR to non-VR. We did not use this kind of control group because bringing tutors physically to the school to engage in mathematics tutoring was outside of the scope of our project. Thus, our analyses cannot assess the unique contribution of VR to the outcomes of the tutoring process—rather, it assesses the effect of mathematics tutoring bundled with VR, as an integrated whole. As we note later, both the experimental and control groups were exposed to the mathematics content during their regular school hours, so both groups had significant exposure to learning activities related to the mathematics content in question. The difference between the groups was the unique contribution of additional VR mathematics tutoring.

3.2 Measures and analysis

We distributed a mathematics pre-test and post-test to all students. These tests had 15 items (7th grade) and 13 items (8th grade). These tests had 8–9 items from the open education resource Illustrative Mathematics (IM; the curriculum many of the simulations were based on) assessments, in addition to 5–6 open-ended items asking students to define terms like “prism” or “rotation.” Appendix 2 describes each item on the test, its source (i.e., an end-of-unit test in Illustrative Mathematics), and how it was scored.

Students also responded to demographic items and took an assessment of their spatial reasoning (SRI; Ramful et al. 2017) on the pre-test. This 15-item test measured spatial ability in terms of three constructs: mental rotation (e.g., rotating a 3D wooden block), spatial visualization (e.g., mental folding of the net of a cube), and spatial orientation (e.g., navigation and point of view). On the post-test, students completed items from the Dynamic Geometry Assessment (DGA; Masters 2010), a standardized geometry test. For 7th grade, we used 7 items from the DGA that examined geometric shapes (e.g., measuring the area of a rectangle). For 8th grade, we used 5 items from the DGA that examined reflection and rotation (e.g., rotating an arrow 180 degrees around a point); items were chosen based on their proximity to the concepts intervened upon. Data were analyzed using OLS regression models with the lm() function in the R software package. Post-test score or DGA was the outcome variable, with pre-test score, SRI, Condition, and other demographic variables as predictors. We first fit the models as multilevel models using the lme4 package in the R software (Bates et al. 2015), using a random intercepts model. However, the nesting variables of tutor, student group, and school math teacher usually had variance components that were very close to zero (variance for “tutor” random intercept < 0.0001 for post-test and for DGA; variance for “student group” random intercept < 0.0001 for post-test and for DGA; variance for “math teacher” random intercept < 0.0001 for post-test). For the DGA outcome, the math teacher nesting variable had a non-zero variance component, so we ran the models as both random intercept multi-level models and OLS regression models to ensure results were the same. The results were the same, so we opted to consistently use a simpler model structure.

The small sample size prohibited looking at interactions. Because both the DGA items and the mathematics pre- and post-test items were different by grade level, for the regression analysis, scores were normalized by grade level—i.e., the mean for each grade level on each assessment was subtracted from each outcome value, and the difference was divided by the standard deviation for that assessment at that grade level. This allowed the tests to be comparable across grade levels despite having different items. There was also a main effect in the models for 7th versus 8th grade. The sample size was 29, as 6 students from the control and 3 from the experimental did not take the post-test due to absence.

We also collected video from each of the tutoring sessions. The 20 students in the experimental condition were organized into 8 groups, which had sessions over 7 weeks (8 × 7 = 56 sessions). However, due to technical issues and absences, there were 46 videos recorded. Our unit of analysis was one video of one 30–60 min tutoring session with one tutor and 1–3 students. All videos were analyzed using qualitative open coding techniques (Strauss and Corbin 2004) to identify the affordances and constraints of VR tutoring. We used the qualitative method of reflexive thematic analysis (Braun and Clarke 2019), in which codes capture interesting features of the data, and themes, patterns of shared meaning, are built from individual codes. For example, we coded specific instances of gesture and movement from the videos and coupled them with timestamps, and then these codes evolved into themes that gave different general types of uses of gestures and movement in the environment. This flexible approach emphasizes transparency, continuous reflection, and assumption-checking. We applied thematic analysis as a contextualist method, acknowledging how our participants make meaning of their experiences and how the broader social context contributes to those meanings (Braun and Clarke 2006). The entire dataset was coded by 5 coders. Throughout the analysis process, the authors worked independently and collaboratively. During independent work, the authors viewed the video data corpus, flagged ‘hot spots’ of activity (Jordan and Henderson 1995), and generated lists of possible coding categories. During collaborative work, the authors engaged in ‘video clubs’ (Jordan and Henderson 1995) where they shared parts of the video data they found interesting, discussed the emerging coding categories, and refined these categories into a set of final codes. Through these discussions, a final codebook was developed (Appendix 3). Then, using the new codebook, the dataset was re-coded by a sixth coder, to obtain the themes in Table 2. We did not calculate inter-rater reliability, as the themes in Table 2 were not modelled quantitatively, nor were the count of instances of each theme intended to be directly interpreted numerically. Inter-rater agreement figures can oversimplify a complex process and focus too much on the endpoint (agreement) rather than the process (discussion). Instead, we take a critical approach to validity which acknowledges that knowledge construction is fundamentally a social and cultural enterprise, and therefore, imbued with the social positions and interests of the knowledge constructors (e.g., Fairclough 2003). As five coders with diverse backgrounds across education, mathematics, technology, and VR, we integrated our own perspectives into the coding process to develop codes situated in theories of embodied learning, mathematical play, and spatial reasoning, treating our backgrounds as a resource rather than an impediment to objectivity. We systematically immersed ourselves in the video data, watching videos multiple times while considering different lenses over an extended period of time. We engaged in collaborative coding to increase reflexivity. Our themes do not capture every facet of the data, but rather emerge at the intersection of the data and the researcher engagement with the data. For example, the mathematical play was not an element we had considered prior to the study or purposefully designed into the environment, but our interactions with the data progressively highlighted its importance.

Table 2 Qualitative themes generated from video data
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Multimodal analysis, a technique for analyzing interactions rooted in embodied cognition theories (Walkington et al. 2024c) was used to more deeply analyze multiple instances of each theme, and selected excerpts from the multimodal analysis were abbreviated and included in the paper. To demonstrate the validity of our interpretations, we provide as much data as possible throughout our findings to guide the reader through our analytical process. This allows our readers to see what aspects of the data we coded and determine for themselves whether they believe our interpretations are valid. Furthermore, throughout data collection and during analysis, researchers consulted with the VR teachers to member check the emerging results.

4 Results

4.1 RQ1: the effects of VR tutoring on math achievement

Descriptive statistics are shown in Table 3. The table shows the average scores (with standard deviations) for the experimental and control groups on the pre- and post-test, as well as average scores on the SRI and DGA.

Table 3 Descriptive statistics
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Regression results are in Table 4. The Model 1 panel of the table gives regression results for the post-test with the Illustrative Mathematics items, while the Model 2 panel of the table gives regression results for the standardized Dynamic Geometry Assessment administered at post. The tables show the regression coefficients for the outcome variables, which are standardized by grade level, as well as the error of the regression coefficients and their significance level. Both models control for gender, grade level, pre-test score, school site, and spatial reasoning score.

Table 4 Regression models predicting post-test scores
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On the IM post-test, the VR math tutoring group significantly outperformed the control group, with an estimated performance difference of 1.07 standard deviations (p = 0.0054, 95% CI [0.39, 1.75]). On the DGA post-test, the VR math tutoring group significantly outperformed the control group, with an estimated performance difference that was 0.77 standard deviations (p = 0.0227, 95% CI [0.15, 1.39]). Appendix 2 gives a detailed breakdown of how students in 7th and 8th grade, in the control and experimental groups, scored for each item on the IM pre- and post-tests.

Although these results for the tutoring intervention are promising, our quantitative analysis was limited due to a small sample size, a weak control group (i.e., students who participated in non-math VR activities), and the administration of the DGA only post-intervention. Thus, these quantitative results should be considered exploratory only, and are presented mainly to provide context for the qualitative results we present next. However, it should be noted that both the experimental and control groups were learning about the respective math topics (prisms and pyramids, motion geometry) in their schools during the tutoring intervention period, and it is notable that the experimental group may have been able to further improve school learning through the tutoring.

4.2 RQ2a: novel affordances of VR math tutoring

We identified 4 themes for affordances of VR math tutoring and 3 for constraints of VR math tutoring, and we first detail our 4 affordances.

4.2.1 Affordance 1. Student whole body movements

Students used whole body movements (or the teleportation feature) to travel around and within mathematical shapes; this occurred in 42 of the 46 videos. Many of these interactions could not have happened in either the real world or in a typical online tutoring context. Figure 1 shows examples. In the top panel (Fig. 1a), the tutor asked the students to make the rectangular prisms big enough to fill the whole room, and then asked them to move their virtual bodies to join him in the middle of the inside of the prism. Once there, the tutor asks the students to look around and make observations. This kind of embodied perspective-taking may allow for understanding mathematical figures in new ways. In the middle panel (Fig. 1b), we see a student Tiger bending his body down and backward to see the cube from below. The student wanted to see if there were missing pieces in the middle of the cube that could be viewed from this new visual, 3D perspective. This body movement allowed the student to shift what was perceptible about the shape, potentially giving a more complete understanding of its properties and allowing for generative learning. In Fig. 1c, the tutor asked students Cherry and Golden where they think the triangle will be when it is rotated (left). The tutor asks them to move their virtual bodies to the position where the triangle will be (middle), and then shows them where the triangle actually is with that rotation (right), adding on a coordinate plane grid. Here, the students could coordinate their embodied gestures and movements with the movement of the virtual object, allowing for the possibility of new perceptual-motor schemes. They also could compare and contrast their embodied reasoning with that of a partner, allowing for new ideas to surface through embodied collaboration.

Fig. 1
figure 1

Students use whole-body movements to navigate mathematical aspects of the environment

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Generally, for larger movements around the space, the students had to use the teleportation feature (e.g., Fig. 1a), whereas for smaller movements, often to get different perspectives (e.g., Fig. 1b), students would move their actual body. Each of these types of movement strategies likely has different affordances and constraints relating to students’ sense of embodiment and ability to abstract sensorimotor patterns from their movements. Teleportation allows students to instantly take on dramatically different embodied perspectives on mathematical objects. However, the use of such a feature may impair sense of embodiment.

4.2.2 Affordance 2. Dimensionality

The dimensionality theme was found in 30 of the 46 videos. Students could interact with shapes in a way that was uniquely three-dimensional, and many of these interactions could not have happened in either the real world or in a traditional online tutoring context. In Fig. 1a, the students are doing a uniquely three-dimensional task where they have to fill a room with a rectangular prism that grows in three dimensions as it is pulled. This activity would be impossible in the real world, as modifying the size of a prism by stretching it with your hands would be impossible, and the scale of the prism would quickly become unwieldy. In Fig. 1b, students interact with the cubes in a uniquely 3D way—putting them beside each other in three directions (x-, y-, and z-), and observing how they look different from different 3D perspectives. This may help make important connections between interacting in three dimensions with objects in the real world and grasping abstract mathematical principles in the VR environment. The tutoring environment also gave rise to many explicit discussions about dimensionality and the difference between 1-, 2-, and 3-dimensions. An example is shown in Fig. 2, where Penguin uses gestures and body movements to show how the shape he is working with can look 2D from some visual perspectives, but 3D from others. His embodied gestures and body movements may allow for his cognition to be offloaded onto the environment, and for others in the environment to better grasp his ideas. Taking different perspectives can also help students understand how their collaborators might differently view and interpret mathematical objects based on their different physical location (e.g., see Washington et al. 2024).

Fig. 2
figure 2

Penguin and the tutor discussion dimensionality

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4.2.3 Affordance 3. Collaborative mathematical play

Another affordance of the VR environment is that students could engage in mathematical play, which included theatrical or game-based ways to interact with the environment, their virtual bodies, and the mathematical objects. This happened in 20 of the 46 videos, and much of the play seemed to arise due to the “fun” and imaginary elements of the VR environment, and the fact that multiple students were in the environment together and collaborating. In Fig. 3, students are playing the game “Simon Says” with the tutor. The tutor first asks them to raise their left arm (Fig. 3a, left) followed by their right arm (Fig. 3a, right), and then asks them to rotate their bodies by 360 degrees (Fig. 3b). Such games are unique to the VR environment because students can engage in motions like translations, reflections, and rotations using their whole body while positioned within a room-scale 3D coordinate plane. They can also collaboratively discuss and evaluate the embodied movements of their peers in real time, updating their own perceptual motor schemes for different math concepts.

Fig. 3
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Example of students engaging in tutor-initiated collaborative mathematical play

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While some instances of mathematical play were tutor-initiated, others were driven by the students. In Fig. 4, the student decided to build a dinosaur with the cubic blocks, and then tested mathematically rotating the dinosaur in different ways to see how they could make it look like it was walking. It is interesting and playful that gravity would disallow the students’ design in the real world (i.e., the blocks would fall). However, this instance of embodiment may allow students to map between math concepts and a playful representation of a real-world being. Other examples included students collaborating to playfully reflect each other’s virtual body movements in a “mirror” and students pretending to have lightsabers that could slice through the 3D shapes to transform them.

Fig. 4
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Example of student-initiated collaborative mathematical play

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4.2.4 Affordance 4. Tutor and student gestures for reasoning and collaboration

A final affordance is that teachers and students could use hand gestures, including iconic gestures and pointing gestures, to communicate ideas to each other and advance their understanding. In traditional online tutoring environments, communication via gestures is often limited. Students made gestures in 41 of the 46 videos and tutors made gestures in 43 of the 46 videos. Figure 5 shows examples of student and tutor gestures. In Fig. 5a, the tutor is demonstrating the mathematical concept of reflection by (left) holding her hands with palms facing upward in the XY plane, (middle) lifting her left hand as if it is popping out of the XY plane, and (right) flipping to the other side of an imaginary line of reflection. In Fig. 5b, the tutor and the student Cbus enact a translation together by (left) stacking their hands on top of each other, and (right) sliding the stacked hands away from each other. Both are instances of iconic gestures that represent mathematical concepts, and the second is a collaborative gesture. In Fig. 5c (left), Penguin uses a pointing gesture to indicate the apex of a cone, and in the right panel, Cat and Jo point to and count the sides of the virtual object. Gestures can both reflect mental simulations and give students ways to offload cognition onto the body and make new connections. Observing and building on each others’ gestures and actions collaboratively can be especially important to advance mathematical reasoning, including when this collaboration occurs between students and tutors. Beyond gestures, the VR environment also allowed students to enact other bodily cues to show attention and give responses. Figure 5d (left) shows a student making a “thumbs up” gesture to show they understand, while Fig. 5d (right) shows a student turning around to look at the tutor. Tutors would also periodically ask students to create their own hand gestures to correspond to mathematical vocabulary words. In Fig. 6, we see two students (Cat’s and Tiger’s) response when the tutor asked them to make perpendicular lines with their hands. This unique activity can allow students to compare and learn from each others’ gestures in a more explicitly-structured way.

Fig. 5
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Students and tutors using hand gestures for reasoning and communication

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Fig. 6
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Students instructed by the tutor to make a gesture showing perpendicular lines

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4.3 RQ2b: novel constraints of VR tutoring

Several of the constraints of VR math tutoring are obvious and do not make sense to explore from video footage. There is the expense, setup, and expertise involved in using VR. However, we identified three additional constraints from our videos.

4.3.1 Constraint 1. Connection problems

One constraint of the VR tutoring environment was regular issues with the students experiencing connection problems; this happened in 21 of 46 videos. We found that having multiple people in different locations as avatars in a virtual room with dynamic virtual objects was complex technically, and crashes could be frequent. Note that this occurred even though the research team provided our own mobile hotspots at the school, rather than relying on school internet. These crashes could also interrupt collaboration if some students were disconnected while others were not.

4.3.2 Constraint 2. Perspective/orientation

A second constraint of VR for math tutoring was the challenges involved with perspective and orientation in 3D space. These issues were in 19 of the 46 videos. In the VR environment, figuring out which quadrant was the “first quadrant” or which direction was “counter-clockwise” was dependent on the way each person’s virtual body was oriented—so it could be different for each person. This could both necessitate novel forms of collaborative conversation and gesture, and disrupt collaboration from being able to effectively take place if negotiation of spatial norms was too difficult. Students also had to negotiate which axis was which in the coordinate plane, such that they could successfully move objects in the x- or y- direction. In Fig. 7, we see the tutor working with Golden and Error to go over which quadrant is which. The quadrants and axes are not labelled, so the students and the tutor must use pointing gestures and bodily positions to collaboratively establish shared meaning around how they will reference the quadrants. In Fig. 8, we see the tutor in the VR environment with 3 students (Geo, Dio, and Cherry) discussing how “her clockwise” rotation is different from “their clockwise” rotation. Geo, Dio, and then the teacher each make “clockwise” sweeping gestures with their arms to show what clockwise means from their perspective. The tutor works to establish a shared understanding of why their rotational directions are different, closing the transcript with “So that means we need to maintain perspective, right?” While we coded this as a constraint, as it caused students confusion, allowed them to make mistakes, and took additional instructional time to negotiate, it could also be seen as an affordance. Having to negotiate these mathematical conventions to create shared meaning could deepen students’ understanding of why we mathematically navigate the world in this way, how conventions are formed and what they mean, and could increase their engagement and investment in the VR environment. It also offers novel opportunities to launch discussions that would only be possible if multiple bodies were in a virtual environment simultaneously, trying to navigate their different positions and perspectives.

Fig. 7
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Example of establishing a shared perspective when orienting to quadrants within a coordinate plane

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Fig. 8
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Example of issues with direction of rotation on the coordinate plane

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4.3.3 Constraint 3. Off-task or distracted students

The VR environment would also cause distractions—like students playing with glitchy aspects of their avatars or play-fighting. Off-task learners were coded in 25 of the 46 videos. Students would walk around, play with their virtual hands (e.g., play rock/paper scissors), climb the virtual tree outside the classroom, or spin around in the environment using a rolling chair outside the environment. These kinds of actions were playful, and in our view, not necessarily undesirable. Students do not need to be engaged in mathematics learning every moment and being able to take fun and often social/recreational breaks is important for rapport in learning environments. Such rapport may help students be motivated to work and think together collaboratively when confronted with mathematical tasks.

There were also instances that were less playful—where learners were talking to someone or on their phones in the real world. Body language was more limited in VR than in the real world, and tutors could not always easily tell if the student was doing something like texting on their phone or eating in the real world. In Fig. 9, the tutor saw one of the students in the simulation, Cherry, moving around in the VR space and moving her lips like she was talking, but the tutor could not hear her voice. After the transcript ended, it was confirmed that Cherry’s headset was not muted or malfunctioning, and that she was likely choosing not to engage in the VR environment or speaking very quietly. The tutor’s lack of access to her real body with its body language limited his ability to diagnose the issue and led to a lengthy process where Cherry had to be located in the real world. This could interrupt both collaboration and the ability of tutors to move the instruction forwards.

Fig. 9
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Tutor is unsure from Cherry’s virtual body language if she is participating in the tutoring

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5 Discussion and implications

We examined a VR-based tutoring intervention for middle school mathematics. Leveraging theories of embodied cognition (e.g., Wilson 2002), we describe how students gestured, moved, and otherwise engaged in an embodied manner with their virtual bodies to learn mathematics from human tutors. The VR environment was collaborative, such that the tutor was in a virtual room with a small group of students, with dynamic mathematical objects (Hollebrands 2007) that could be manipulated and explored. This work is significant because it is the first attempt we are aware of to study mathematics tutoring in VR. Through our analyses, we identify important topics for future design in an area that is critical to recovery from the COVID-19 pandemic. We also show the potential of tutoring activities specifically designed with theories of collaborative, embodied learning, building from a tutoring literature base that does not consider these important elements of human learning. Our small sample size ultimately limited the implications we could draw from our first research question, so the main contribution of this study is in our second research question and the rich, qualitative analysis of video data from the VR sessions, which is not typical in many other studies of math learning in VR.

5.1 Research question 1

Our first research question asked what the impact of the tutoring intervention was on students’ math achievement, with respect to students in a control group who did activities in VR that were not math-related. Our quantitative analyses found some evidence of promise, but our results are exploratory only given the small sample size and weak control condition. Looking at results for specific items on the posttest (Appendix 2), we see mixed results. The VR math group showed strong increases compared to the control on some of the definition questions (questions 4 and 5 in 7th grade on “surface area” and “base”; questions 4 and 13 in 8th grade on “dilation”), but they did not score strongly when asked to define a “rotation” or “reflection,” showing slight decreases from pre- to post-. The VR environment likely forefronts some math principles while deemphasizing others, and key elements of concepts like rotation and reflection may have been left out of the tutoring conversations (e.g., students were slightly more likely to explain rotation as a turn or spin without elaborating on the post-test). In 8th grade, however, we saw some of the strongest performance of the VR group compared to the control group on items where students were asked to apply the concepts on rotation and reflection to specific figures, suggesting that their ability to apply these concepts may have still been enhanced. In 7th grade, VR math students made the greatest gains with respect to the control group on items about pyramids, but showed weakest performance compared to the control group on items about prisms. It might be that the VR math environment was more effective for principles where students had less general prior knowledge (e.g., pyramids), whereas they had been working with prisms since 5th or 6th grade.

Our theoretical framework, provided earlier, gives some indications as to why these learning gains may have occurred. We posited that the VR tasks would allow for embodied generative learning (Castro-Alonso et al. 2024), where students are able to offload cognitive demands onto their bodies through gestures and actions, and onto their environment through interactions and manipulations of virtual objects. This offloading should allow for more focus on learning the recurring sensorimotor patterns (Abrahamson et al. 2021) that are made visible through interactions with the objects in the environment, as well as through students’ and tutors’ movements and gestures. The environment and interactions within also allowed for important connections to be made to students’ prior knowledge and experience interacting in embodied ways with the world, and such connections can promote learning by bridging relatively abstract mathematical ideas (Nathan 2021). Finally, the collaborative elements of the environment allowed students to observe, reflect on, and gesturally respond to each others’ actions, which can both facilitate communication and understanding, and allow new mathematical ideas to arise and be refined (Walkington et al. 2019). And even though the post-assessments were in written format, the underlying mathematical patterns that students grasped from the environment were present in this different form. Given the prolonged access to the environment and its interactions, and the focus of the tutors on mathematical terminology and progressive formalization, students may emerge with sufficiently flexible embodied understandings to still recognize them on a written post-test. Without the tutors as constant interlocutors, facilitators, and translators, students seeing the connections between their embodied interactions in VR and a written mathematical test may have been significantly more difficult.

Despite this strong theory, going into the study, we were not necessarily expecting to see positive results. This was because of the novelty of the tutoring platform and its relative lack of stability (e.g., connection issues were experienced in 46% of sessions), the untested nature of our embodied tutoring activity design, the issues with student attendance we knew we would be likely to have in an afterschool context (i.e., only 54% of students attended each session on average), as well as the less-than-impressive results in the literature for distance math tutoring (Sass and Ali 2023; Schueler and Rodriguez-Segura, 2023; Zhang et al., 2023; Zydney and Hord 2023). We also knew students would be learning these same concepts in school during the seven weeks, and that we would have to show value-added over this school-based learning. We were surprised to see some initial promising outcomes on the curriculum-based math assessment and on the standardized assessment in our quantitative analyses. This builds on other literature suggesting that distance math tutoring can be effective (Gortzar et al., 2024) as well as literature suggesting that embodied learning can take place successfully at a distance (Shvarts and van Helden, 2023)—but uniquely brings these two strands together. It also builds on research showing the potential of VR activities in mathematics for student learning (Çakıroğlu et al. 2023; Demitriadou et al. 2020; Su et al. 2022).

5.2 Research question 2

Our second research question about the affordances and constraints of VR math tutoring was answered by examining our video data of student-tutor interactions. Our qualitative coding uncovered many practices in the videos that took full advantage of the fact that students and tutors had virtual bodies and a virtual space to move around in, as well as others to embody concepts with—including regular use of whole-body movements and gestures, in distinctly mathematical ways, both to reason and to communicate. This extends research on embodied learning in mathematics (e.g., Castro-Alonso et al. 2024), and collaborative embodied learning in mathematics (Walkington et al. 2019), by showing how virtual bodies can collaboratively embody math concepts together. And this prior research suggests that these interactions can both improve collaborative communication and give students new ideas about mathematics, which can both improve learning. Our themes showed moments of students both demonstrating and building their mathematical knowledge through embodied resources like gestures, actions, and movements. This also builds on research on gestures (Alibali and Nathan 2012; Hostetter and Alibali 2019), by showing how students gesture using virtual bodies, and the limitations that these virtual bodies have in terms of body language for understanding students’ participation in collaborative contexts.

Our qualitative coding also revealed how this embodiment students engaged in often became playful, in both mathematical and non-mathematical ways. We thus extend research on mathematical play (Williams-Pierce and Thevenow-Harrison 2021) by showing forms of play that arise when students have access to virtual bodies in a collaborative space with unique embodied affordances. To our knowledge, little research has connected math tutoring to mathematical play—indeed, these two topics seem almost antithetical to each other. Our work, however, suggests that play can incorporate math concepts that can enhance tutoring and the embodied interactions therein, and that connecting play and tutoring may be beneficial.

Further, our qualitative coding showed how the playfulness that students were able to engage in was often driven by the dynamic mathematical objects that they had access to—objects that they could manipulate with their hands, and that they could see updated in real-time as their avatars and others engaged in manipulations (Maravita and Iriki 2004; Chandrasekharan 2009). These kinds of collaborative DGS-type environments may be the future of mathematics tutoring. Here we extend research on DGS (Hegedus and Otálora 2023; Hollebrands 2007; Leung et al. 2013; Pittalis and Drijvers 2023) by giving accounts of students using this DGS in uniquely three-dimensional ways (Dimmel and Bock 2019), such that the DGS had brand new affordances (e.g., filling the room you are in using a dynamic rectangular prism). We also provide caution for a new issue that arises with DGS in VR—negotiating issues of perspective and orientation when different people’s virtual bodies are positioned differently.

Finally, our coding revealed issues with students orienting themselves and their bodies in the VR environments. There are several possible explanations for this issue. The simplest is that students would have had the same issues in the real world, if they were placed with their real bodies on a large, physical coordinate plane that was unlabeled. Learning to navigate in a Cartesian coordinate system in general requires new thinking about spatial ideas, regardless of whether the environment is virtual or real. However, there were also probably other factors compounding this issue. Students have individual differences in their spatial reasoning abilities, which can influence the degree to which they can move, navigate, and interact successfully in VR (Gittinger and Wiesche, 2024). In addition, students’ sense of embodiment (Kilteni et al. 2012) in the VR environment might be variable, and students with higher body awareness may have more difficulty orienting themselves in VR than they would in the real world (Chatain et al. 2024). The fact that the avatars lacked legs and that there was a somewhat limited selection of possible avatars to choose from might have also degraded students’ sense of body ownership, which already may have been negatively impacted for students with high body awareness (Chatain et al. 2022). Future research should examine how sense of embodiment and spatial ability measures relate to issues with mathematical perspective and orientation in VR environments.

5.3 Recommendations for VR-based math tutoring and future research

We close the article by providing five recommendations for VR-based math tutoring. These recommendations are derived primarily from our qualitative analysis of the video data, and any such design recommendations will ultimately need to be verified in rigorous experimental studies.

  1. (1)

    Design VR tutoring such that it is collaborative, with multiple students in a virtual room with a tutor but be aware of the technical issues this may cause with having a reliable connection for all users.

  2. (2)

    Encourage the tutor and the students to use hand gestures and whole-body movements to explore the virtual world and the objects in it mathematically. Have students constantly use and think about the position of their virtual bodies and virtual hands in novel ways, with respect to the environment and its objects.

  3. (3)

    Encourage mathematical play during tutoring and have breaks where students can also engage in non-mathematical play. Elements of the environment that are unique to VR and impossible in the real world (e.g., being able to teleport) may be effective to drive play.

  4. (4)

    Consider perspective and orientation issues when engaging in mathematics in 3D space and have a plan to actively address these issues. Also have environmental tools prepared, such as options to toggle on/off labels for axes, options to change the orientation of the coordinate plane, etc.

  5. (5)

    Use VR mathematics tutoring for concepts that are uniquely 3D, and that can leverage the 3D nature of the immersive environment.

Future studies should compare VR tutoring to typical online tutoring and typical face-to-face tutoring, to see what the effect the novel affordances of VR have on student learning. Future qualitative work should explore issues of perspective and orientation in VR, the power of mathematical play in VR, and the nature and opportunities in VR relating to gestures and actions on dynamic objects. As technology continues to advance, we believe VR tutoring will become commonplace, given its affordances for embodied learning, dynamic interaction, and collaboration.