The MOVES-NL utilizes both a body-scale walking NL *and* a small desk-scale NL as an intended means for students to ground the targeted mathematical procedures through blending perceptual perspectives. First, students walk along the body-scale floor-based NL to solve integer arithmetic problems by enacting them (see Figure 1). Students are instructed to: (a) start by standing on the first number in the problem (not shown in the figure); (b) turn to the right (positive side of the NL) for addition problems (see “addition” sign on the classroom wall) or turn to the left (negative side of the NL) for subtraction (see “subtraction” sign on the classroom wall); and (c) walk the amount of steps indicated by the second number in the problem (forwards if the number is positive, backwards if the number is negative). Figure 1 exemplifies enacted solution moves for the four possible combination schemes of adding or subtracting (columns) positive or negative integers (rows) on the walking NL as modeled by the teacher’s three-move instructions (see also Anton & Abrahamson, under review). Note here that students are experiencing the NL from an egocentric perspective (Tversky & Hard, 2009), whereby the NL is positioned on the sagittal (front–back) axis in respect to the body.

Next, students are invited to sit at their desks. They are offered a tablet-based NL as well as a figurine. They are asked to use this action figure to reenact their own body-scale arithmetic operation moves, now at desk-scale. Note here that, in this case, students are experiencing the NL from an allocentric (third-person) perspective (Herbst et al., 2017) even as the figurine “experiences” the NL from an egocentric perspective. Movement along a sagittal axis has been shown to prioritize an egocentric perspective, while lateral movement prioritizes an allocentric one (Margetis et al., 2020). We conjecture that having students experience the NL from both an egocentric and (by surrogate proxy) allocentric perspective will facilitate the form of perspectival coordination that students require in order to make sense of the disciplinarily normative desk-scale NL in terms of their enactment on the body-scale NL; and that walking the action figure along *its* egocentric pathway even while seeing it from an allocentric perspective will create necessary cognitive circumstances for a phenomenological blending of the perspectives (e.g., as when we learn to operate a car or a comb from mirror images). We are intrigued by the cognitive mechanisms, challenges, and opportunities, of thus splitting and synergizing sensorimotor perspectives, where the eyes are *seeing* the NL while our operating hand is *being* the NL (cf. Gerofsky, 2011) as well as by the conceptual prospects of this perspectival complementarity (Abrahamson & Bakker, 2016; Benally et al., 2022).

## MOVES

This educational design utilizes MOVES to create three different NL-based interactions. The first interaction (Figure 2) includes a NL ranging from –5 to +5 projected onto the floor along with either an addition or a subtraction problem projected onto the wall. The dual wall-and-floor projectors are coordinated through the *SENSEi* software (Gelsomini, 2023), which allows the motion sensor to track students’ position, orientation, and movement on the NL and, in response, mark the current position dynamically on the floor projection. In particular, when students stand on each hash mark along the NL, the number under their feet turns blue and a pleasant chime is sounded. This way, students can see and hear that the motion sensor is capturing their position. SENSEi’s interactive sonification affordances are particularly important for blind and visually impaired student accessibility, albeit the current paper will not elaborate on the potential inclusivity parameters of future variants on MOVE-NL that will cater to sensorimotor diverse students.

In addition to recognizing student *position* on the NL, the projector registers student *orientation* and *movement*. As the student performs the correct movements, the problem on the wall lights up in green and a congratulatory sound is played, providing students with in-the-moment feedback on their whole body movements. For example, given the problem “ - 1 – 2,” the student would first stand on the NL’s -1 hash mark. As they do so, the -1 on the floor-projected NL lights up in blue, and a chime is played. Concurrently, the -1 on the wall-projected NL in front of the student lights up in green. Next, the student needs to turn to the left, in order to orient themselves in the subtraction direction (still before moving). As soon as the student turns left, the subtraction sign on the wall turns green. Finally, the student needs to take 2 steps forward (i.e., in the direction they are facing, which is toward the lesser values on the NL). Once the student has taken the 2 steps, they raise their hands in the air to signal that they have reached the solution. If the solution is correct, the entire problem on the wall is highlighted in green, a congratulatory sound plays, and the solution is displayed.

The second interaction is largely the same as the first, only that a virtual avatar projected onto the wall mirrors the students’ position and movements on the walking NL. See Figure 3 for an illustration of the second interaction.

Here, the student receives the same feedback from the motion sensor as in the floor-only earlier activity. In addition, however, the avatar projected onto the wall mimics student movement. The design rationale of deploying a mirrored avatar in full view of the student is to support the student in bridging the egocentric experience of walking along the NL with the allocentric experience that is required in the final interaction, when the student is seated at a desk.

The third and final interaction involves only a tablet, which displays a smaller, desk-scale NL and, again, presents an addition or subtraction problem (see Figure 4). During this interaction, the student reenacts their previous whole-body movements by moving a tangible figurine (of identical appearance as the virtual avatar) along the small NL, just as they had moved their whole bodies on the walking NL.

Similarly to the previous levels of interaction, the tablet recognizes *where* the student places the figurine, in what *direction* the figurine is facing, and what *steps* the figurine is taking. When the student places the figurine in the correct location and facing the correct direction, the various corresponding screen elements of the displayed problem are highlighted in green and a congratulatory sound plays. In Figure 4, the student has correctly completed the first phases of solving “-1 – 2 = ?” (begin by standing on -1; note that he has not yet performed the second phase of facing the avatar toward the lesser NL values per the item’s subtraction operation symbol).

## The MOVES Technological System

The hardware structure of *MOVES* (Cosentino et al., 2023) is both solid and flexible (see Figure 5). Its base has wheels (C) that can be locked for the duration of the activity yet allow for easy repositioning and transport per diverse environments. The platform holds a mini-PC (E) that reads motion-sensing (RGB and depth) video and audio streams (Orbecc Astra Pro) (F) and outputs to two Ultra Short Throw LED projectors (G) using two independent video outputs (projecting the NL interactive image on the floor as well as the arithmetic problem on the wall). A router (I) provides wired connectivity to the PC, creating a local Wi-Fi network to which a controlling device (smartphone, tablet, or remote controller; L, M) is connected to control the experience. A coat made of a PVC layer covers the structure’s front. The MOVES platform is equipped with SENSEi software (Gelsomini, 2023). *SENSEi* is a suite of software modules that enables the PC to manage several input and output devices. At a lower level, these devices are recognized and communicate with the PC through the use of traditional drivers. At a higher level, they are accessible in the form of simple, homogeneous, and intuitive APIs with which novice-to-skilled programmers can interface. SENSEi enables this simplification by accessing device providers’ Software Development Kits (SDK) and translating them into a standardized documented form. The software is installed as a set of modules that interface with the sensing and actuation devices and a viewable layer to which contents are displayed.

## Pilot Study

A pilot study using the MOVES-NL was conducted with twenty students in Grades 4–6 (9–11 years old) in Milan, Italy. (Note that, by this age, students will have studied basic arithmetic with negative numbers, albeit their understandings, per the literature, would not be robust at best.) Students began with the walking NL interaction (Figure 2) and then operated the walking NL with a mirrored avatar (Figure 3). Finally, students solved problems using the tablet (Figure 4). Throughout the interactions, the researcher collected various observations and engaged students in a semi-structured interview (Ginsburg, 1997), seeking to gain deeper understanding of students’ conceptual processes. All interactions were video recorded.

Analyzing these data, we observed that students’ implicit confusions around negative integer arithmetic emerged as they were asked to represent procedures and solutions through whole-body movement. Students typically completed the first two steps of the walking NL correctly (stand on the first number; face either addition or subtraction). However, students hesitated when it came time to take a step, often wondering in which direction they should walk. This hesitation is a case of misinterpreting the contextual function of a semiotic resource, here the actionable meaning of the polarity of the second number (i.e., whether it is positive or negative). This finding serves as evidence supporting a claim that whereas children have enacted arithmetic operations throughout their childhood, they have not had the opportunity to enact numerical polarity (Mock et al., 2019).

Furthermore, students’ hesitation to step either backwards or forwards could reflect the polysemy of the “-” sign. This “-” sign can either be operational in nature (i.e., subtraction) or it can denote polarity (i.e., negative). The walking NL asks students to address this symbolic ambiguity (Foster, 2011) by stipulating an action-based semiotic differentiation between the operation of subtraction and the polarity of a negative number. However, symbol polysemy is prevalent in mathematics, and students experience tension between the “obvious” well-known symbolic meaning and alternate meanings (Mamolo, 2010, p. 249). In this case, the “obvious” meaning of the “-” sign is subtraction, or “to take away.” The tension arises when students encounter this sign *after* another operational sign (i.e. 1 + –3, or 1 – –3). Notwithstanding, this ambiguity *as evidenced in publicly displayed bodily enactment* provides an opportunity for students and teachers to engage in productive discourse around mathematical concepts (Abrahamson et al., 2009; Foster, 2011). In this instance, we believe that it is imperative for the teacher/researcher to step in, literally, and provide some context, usually by asking guiding questions (Ginsburg, 1997) that will elicit a productive negotiation toward a common understanding of why students should walk backwards or forwards.

After their first interaction, students watched an alien avatar mimic their whole body movements (see Figure 3). The purpose of this interaction was to facilitate students’ biperspectival coordination between their egocentric experience on the walking NL and, prospectively, their allocentric experience with the tablet (Figure 4). In general, students seamlessly transitioned from the first two walking interactions to the tablet, where they mimicked their whole-body movement by operating the avatar “mini-me” action figure. These students were able to coordinate their perspectives to achieve either perspectival mutuality or synergy (see Benally et al., 2022). However, this perspectival coordination was, at times, brief; when using the tablet to solve problems that were similar to those they had already solved on the walking NL, students occasionally reverted to past “school” strategies and thus became “stuck.” It appears that when students are working on blending perspectives in the service of mathematical learning, they need to be given the opportunity to move back and forth between the enactive (walking NL), iconic (mirrored avatar), and symbolic (tablet) interactions (Dutton, 2018; cf., Bruner, 1966).