Mathematics MOVES Me: Digital Solutions for Coordinating Enactive and Symbolic Perspectives—The Case of Basic Arithmetic With Positive and Negative Integers

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Mathematics MOVES Me: Digital Solutions for Coordinating Enactive and Symbolic Perspectives—The Case of Basic Arithmetic With Positive and Negative Integers

https://doi.org/10.21203/rs.3.rs-3597593/v1

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We present an innovative educational design for basic arithmetic that responds to students’ documented difficulties with adding and subtracting single-digit positive and negative numbers. The design utilizes MOVES, a technological architecture that combines floor- and wall-projected interactive interfaces. Students enact arithmetic operations, e.g., “3 - (-2)” by walking along a projected body-scale number line, while their actions are captured and analyzed to provide in-the-moment feedback on elements of their solution procedure. Next, a screen-based avatar is introduced who mimics their whole-body movements. Finally, analogous problems are presented on a tablet that utilizes tangible interaction, where students walk the avatar, now as an action-figure, along a standard-sized number line. Our theoretical framework, design conjecture, product evaluation, and data analysis all pertain to fostering conceptual understanding through coordinating full-body egocentric experiences on a body-scale number line with the allocentric experience of “puppeting” the avatar along the desk-scale number line. Based on pilot trials, we speculate on the nature and type of supports students require to coordinate these perspectives and discuss implications for future iterations of the design.

embodiment

integer operations

multisensory interaction

number line

perspective

This snapshot describes an innovative educational design (MOVES–NL) which utilizes the number line (NL) as a semiotic resource for students to learn how to add and subtract positive and negative integers. In this design, students experiment with interactive floor- and wall projections of NLs generated by the MOVES multisensory technological system, which we describe below.

Relevance for Mathematics Teaching and Learning

Foundational mathematics skills are important predictors of secondary and postsecondary success for students (Varma & Schwartz, 2011). In early mathematics classrooms, students often struggle when they encounter negative integers (Bossé et al., 2016; Hawthorne et al., 2022). Negative integers, unlike positive integers, are not easily modeled by teachers. For instance, if students learn to count, add, and subtract by manipulating concrete “things” (i.e. fingers, blocks, toys), then negative numbers pose a substantial challenge, because it is unclear how to generate and display negative numbers as things that are available for inspection and enumeration. Still, an understanding of negative integers can plausibly draw on an understanding of positive integers. Indeed, providing students with dedicated semiotic resources facilitates a shift from enacting operations with concrete positive-integer tokens to enacting analog operations with quasi-concrete negative-integer tokens (Varma & Schwartz, 2011). Specifically, the NL is a beneficial pedagogical resource for learning negative-number arithmetic, because it affords conceptual opportunities to assimilate negative numbers as spatially deployed variant elaborations of positive-number ontology (Bossé et al., 2016). The present design attempts to augment the NL’s affordance for learning positive-and-negative integer arithmetic by installing prior designs in interactive digital media that offer supports for perceptual coordination across scale (body-scale vs. desk-scale), while supplementing the activities with conceptually oriented feedback regimens.

Perspectival Coordination

The MOVES–NL design creates conditions for students to experience integer arithmetic on the NL through both an egocentric and allocentric perspective. Previous studies that incorporated activities of enacting arithmetic along a body-scale NL (e.g., Nurnberger-Haag, 2018) did not find significant changes in students’ abilities to later solve addition and subtraction problems when given a traditional paper-and-pencil task. We propose that the issue may have been not a shortcoming in the fundamental rationale of body-scale arithmetic enactment but, rather, that the study participants had little to no opportunities to coordinate their body-scale egocentric (first-person) experience of walking along a floor-based NL with the allocentric (third-person) perspective of looking at a NL from the “outside,” which is typically required in classrooms (see Papert, 2004, on “paper math”). The three interactions described, below, in our design seek to provide supports for students to coordinate egocentric and allocentric perspectives on the NL so as to ground their fluency with the traditional NL in their enacted experience on the floor NL. Building on Benally et al. (2022), we frame this design effort as supporting students in first achieving perspectival mutuality (i.e., using an alternate perspective to inform their own) and, eventually, achieving perspectival synergy (i.e., combining two perspectives into a greater structure). This perspectival synergy would subsist of a linear, spatial–numerical mental NL (Mock et al., 2019) that grounds negative-integer arithmetic in concrete action (Varma & Schwartz, 2011).

Embodied multisensory interactions for learning

Finding new methods to improve and facilitate learning is a key objective in both child–computer interaction and learning-technology research (Giannakos et al., 2020). Several advances in sensing technologies (e.g., size, affordability, ease of use, supporting frameworks) allow human–computer interaction researchers to “sense” and “respond” to the users’ presence as well as their gestures, affective states, motions, and manipulations, while simultaneously orchestrating the interactions in numerous ways. In addition, sensing technologies have been particularly beneficial for enabling children’s play and learning as naturalistic and meaningful (Sharma et al., 2021). In particular, multisensory technologies focus on supporting learners’ needs and developmental progression, with early investigations identifying the benefits of multisensory interaction to support academic learning (Zou et al., 2017). Multisensory technologies are digitally connected, controllable, and interactive, providing children with more affordances and enabling ludic-cum-educational experiences in an organic and embodied manner, all of which are important for children’s learning and development (Hourcade, 2015; Malinverni et al., 2019). For example, Kosmas et al. (2019) investigated how to employ a motion-based embodied learning game to improve students’ memory performance when learning a second language. Technologically enabled embodied interaction activities have proven their potential to enhance children’s learning (Lee-Cultura et al., 2020). In addition, Cosentino et al. (2023) investigated the benefits, challenges, and trade-offs between different interaction modalities (e.g., full body interactions) in the context of educational multisensory environments for children. The interaction modalities presented in the next sections (walking NL, walking NL with mirrored avatar, and small NL with figurine), when orchestrated together, we submit, are especially relevant for learning mathematical content. Taken as a whole, the proposed activity rationale rejects representationalist modes of cognition in favor of enactivist accounts (Abrahamson et al., 2022).

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).

The pilot study led to three general conclusions. First, enacting arithmetic procedures as expansive embodied actions can productively disrupt students’ solution-oriented routines, including their mathematically inappropriate heuristics. Mathematics instruction that prioritizes procedural rules over conceptual underpinnings can undermine students’ understanding, agency, and development of content knowledge and epistemic practices (Erlwanger, 1973; Freudenthal, 1971; Kamii & Dominick, 1988; Nathan, 2012). Instead, activities that step aside from rote procedures, for example whole-body arithmetic enactment, appear to refresh and reground students’ mathematical perceptions in their common-sense situated know-how (Ma, 2016).

Second, if first-person immersive activities, such as those provided by virtual-reality technologies, are to ground students’ conceptual understanding of mathematical concepts, then students should be given opportunities to coordinate these situated egocentric perspectives with allocentric perspectives on the analog symbolic procedures, which are more typically prioritized in classroom learning. More generally, utilizing multisensory technology that enables students to coordinate mathematics skills across multi-perspectival media may promote deeper conceptual understanding.

Third, observing students interact with mathematics content in many different ways may provide insight for teachers and practitioners struggling to convey concepts to their students. In our pilot study, the full-body movement initiated by the MOVES-NL design highlighted student misconceptions that might not have been apparent if we had engaged students in typical paper-and-pencil-based mathematics tasks. By collecting action logs and analytics from various biological sensors (e.g., physiological data from wristbands, skeletal data from motion sensors) researchers may be able to furthermore identify specific moments where students are struggling (e.g., experiencing difficulties in determining the correct orientation or moving slowly due to uncertainty).

Future Work

Future iterations of the MOVES-NL could be enhanced if the system provided further encouragement, guidance, and supports for students struggling to blend their egocentric and allocentric perceptions of body-scale and desk-scale number-line enactments. For example, allowing students to shift back and forth between these interactions, when they are stuck on a problem, could help to facilitate this coordination. In a sense, we are “yes-anding” the received gospel from Jerome Bruner, often articulated as “enactive, iconic, symbolic” (Bruner, 1966) by way of supplementing “...and back again” (Abrahamson et al., 2012; Dutton, 2018). It appears students may sometimes require greater agency and latitude in organizing and pacing their own bilateral coordination between co-signifying semiotic registers grounded in situated enactment (Kaput & West, 1994; Thompson, 2013). One activity form that may occasion students opportunities to blend enactments across scale and perspective is for them to instruct another agent across the registers. For example, a student who is performing subtraction on the desk-scale number line may explain to a peer or an avatar—“walk them through,” so to speak—how to enact the same arithmetic operation on the walking number line, and vice versa. These bilateral bridging activities could potentially help students sustain an enactive grounding of integer arithmetic as they adopt symbolic forms of expression in their mathematics classrooms.

Author’s Contributions

The first and second authors (Jacqueline Anton and Giulia Cosentino) prepared the original draft of the manuscript, and all other authors (Mirko Gelsomini, Kshitij Sharma, Michail N. Giannakos, and Dor Abrahamson) reviewed the manuscript.

Funding Statement

The collaboration between the University of Berkeley, California and the Norwegian University of Science and Technology is funded by the Peder Sather Grant program.

Data Availability Statement

Video and audio data from our study are not publicly available to preserve individual participants’ privacy according to the Committee for the Protection of Human Subjects.

Ethics Statement

All authors confirm that any aspect of the work covered in this manuscript that has involved study participants has been conducted with the ethical approval of all relevant bodies and that such approvals are acknowledged within the manuscript. IRB approval was obtained before the collection of data (protocol ID 2022-10-15703). Written consent to publish potentially identifying information, such as details or the case and photographs, was obtained from the participants and/or their legal guardians.

Conflict of Interest Statement

There are no competing interests.

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No competing interests reported.

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Mathematics MOVES Me: Digital Solutions for Coordinating Enactive and Symbolic Perspectives—The Case of Basic Arithmetic With Positive and Negative Integers

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MOVES

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