Exploring Deaf Aesthetics as Spatial-Geometric Thinking, Acting, and Feeling: A Case Study

Abstract

Spatial skills, while vital to STEM (Science, Technology, Engineering, and Mathematics) and STEAM (Science, Technology, Engineering, Arts, and Mathematics) fields, are fundamental to understanding all mathematics. Yet the absence of spatial development in elementary curricula, particularly geometry, where such skills can be deeply explored, is compounded by a lack of theoretical and empirical research, especially in deaf education, where little research addresses learners’ spatial-geometric understandings and the ways their bodies contribute to developing such understandings. We first review relevant literature to interrelate mathematics, spatial activity, and embodied cognition with aesthetics for STEAM (Science, Technology, Engineering, Arts, and Mathematics) in deaf education. We then present a case study in which we observed and assessed Evan, a deaf student, as he worked on three geometry tasks. This video-based research utilises the Pirie–Kieren Theory/Model to further consider the aesthetic, spatially dynamic, and embodied ways that Evan’s geometric understandings emerged and evolved into more formal mathematical activity. Finally, we discuss the ways the study findings focused on spatial-geometric development support future STEAM education research and classroom mathematics towards growth-oriented learning for deaf students.

1. Introduction

Today in education, research and teaching supporting the development of children’s spatial cognition continues to grow (e.g., Fowler et al., 2022; Technology University Dublin, 2024). This interest can be largely attributed to the demand for STEM (Science, Technology, Engineering, and Mathematics) workplace skills. Spatial skills development is vital for entry into, participation in, and success in STEM-related fields. For example, scientists studying galaxy clusters; computer programmers creating robotic software; mechanical engineers knowing how parts connect and move from multiple perspectives; or a mathematician developing a model that simulates the behaviour of a contagious disease.

Spatial cognition also plays a critical role in science, technology, engineering, arts, and mathematics or STEAM-related fields (Ramey et al., 2020), where the “A” calls for artistic creativity and imagination such as the work of architects in the ecological design of buildings (Al-Jokhadar & Jabi, 2020); landscapers employing landscape patterns (Kong et al., 2022); sound engineers exploring visual and tactile modalities within echolocation (Gao et al., 2020); animation illustrators using Computer-Assisted Design (CAD) (Delahunty et al., 2020); or urban planners designing high-density environments (Munn et al., 2022). For young children, making a castle from Lego bricks, playing an instrument, inventing a dance, cooking, or baking all rely on inherently geometric spatial skills such as orienting, navigating, and moving within different dimensional spaces; manipulating images and objects; as well as perceiving and interpreting events.

Yet fundamental to and inherent in the arts lies another “A” representing aesthetics. Aesthetics can be described as a branch of philosophy concerned with the nature or appreciation of beauty. Aesthetics can also be traced to the Greek term aisthētikós, “pertaining to sensory perception”. This etymological framing of aesthetics provides a starting point for our work. The Greek terms for aisthētikós and aisthánomai indicate the focus on sense-making: “perceive, sense, learn”.

To date, there are few theoretical or empirical accounts that inform how the field might conceptualise, observe, and assess deaf students’ geometric understandings as aesthetic and associated with the embodied aspects of their spatial cognitive development (Healy, 2015; Lang & Pagliaro, 2007; Thom & Hallenbeck, 2021, 2022). We are curious, then, what insights concerning spatial-mathematical processes, skills, and tools might be gained when we attend to deaf children’s aesthetic ways of conceptualising geometry.

Said differently, we are interested in how aesthetics understood as perceiving, sensing, and learning can enable deeper meaning and potential for observing and assessing spatial cognition within STEAM-related areas through mathematics education. For this qualitative case study featuring Evan, a deaf student, we ask: What are the aesthetic ways in which Evan’s spatial activity emerges and evolves as he engages in geometric tasks? What possibilities do Evan’s ways of perceiving, sensing, and learning offer to further understand deaf students’ mathematical growth as embodied cognition?

In this paper, we provide an overview of our rationale for the study and discuss the importance of, as well as the opportunities for, deaf students’ spatial activity when their mathematics are recognised as aesthetic, geometric, and embodied. We introduce the Pirie–Kieren Dynamical Theory and Model for the Growth of Mathematical Understanding (e.g., Pirie & Kieren, 1989, 1992, 1994) to examine these aspects of deaf students’ understanding. We then share the findings from a qualitative study we conducted focused on the spatial actions (Davis & SRSG, 2015) of Evan (deaf) during his Kindergarten and Grade 2 years in mainstream class settings as he worked on three geometric tasks. Using the Pirie–Kieren Theory and Model, we demonstrate its potential for observing, assessing, and evaluating spatial and geometric understandings in aesthetically coherent ways. Informed by the findings of this study, we provide implications and considerations for future deaf education research related to STEAM, methodologies, and language development associated with spatial cognition in mathematics.

2. Literature Review

2.1. Study Rationale

This study, in part, responds to the pervasive reality that language deprivation persists in deaf children despite current efforts to provide them with hearing technology (cochlear implants and hearing aids) or sign language by non-native language models in early childhood (Glickman & Hall, 2019). Degraded linguistic input results in language deprivation, which cannot be simply remediated through the array of services including the use of listening devices, speech and language therapy, or sign language interpreters in later years (Cheng et al., 2019). Permanent cognitive damage is a result of language deprivation and severely impacts academic, social, and emotional development (Glickman & Hall, 2019). Despite urgent calls for early and natural exposure to signed language, nearly 78–85% of deaf students are educated in oral language environments in American and Australian schools (Punch & Hyde, 2010; Shaver et al., 2014). Further still, it is estimated that over 70% of deaf students enter school with a history of inconsistent access to a natural signed or spoken language during early childhood (Grote et al., 2024; Hall et al., 2019).

The lack of access to language is compounded by hearing-centric pedagogies that often emphasise decontextualized learning with non-deaf children who are already fluent in a spoken language upon arrival in school (Howerton-Fox & Falk, 2019). Specialised language development strategies and resources that address or ameliorate the impact of language deprivation are not readily transferable to spoken language classrooms where deaf children are placed according to inclusive education practices adopted by school boards (Anderson & Wolf Craig, 2018). Irreversible language deprivation begins in infancy and persists at epidemic proportions (Hecht, 2019; Caselli et al., 2020). Many efforts to provide sign language interpreting, hearing technologies, and tutorial support provided primarily through itinerant teachers of the deaf or in resource-based programmes designed to support language learning in elementary and high school years (Luft, 2017) do not always effectively address language deprivation. Efforts to address language deprivation often remain stymied or ineffectual due to the often opposing and multiple perspectives of administrators, educators, specialised support personnel, and parents (Weber & Thom, 2026). Language choices remain binarized, often favouring spoken language over sign language. Language deprivation occurs through all language choices as cochlear implant outcomes are highly variable and access to sign language models in early childhood are not always available (Hall et al., 2019; Caselli et al., 2020; Glickman & Hall, 2019).

While language deprivation may look like an intellectual or learning disability, the deaf student may draw upon nonverbal intelligence which, among deaf individuals, is often within the normal range (Mayberry & Kluender, 2018). Of particular note, spatial cognition can be a particular strength of deaf individuals (Mayberry & Kluender, 2018). For instance, visual artist James Castle developed an aesthetic approach that facilitated spatial processes, expertise, and expression (Feinberg, 2023; Beardsley, 2021) reflected in his perspective drawings.

Knowing that spatial cognition is essential for all STEAM areas and geometry is the study of spatial concepts (see Section 2.2), we are interested in how mathematics, when spatially oriented, can engage and benefit deaf children.

2.2. Aesthetics and Mathematics for Spatial Cognition Within STEAM Education

Aesthetics is not to be confused with arts-integration strategies in teaching. Rather, aesthetics is the philosophical study of the nature or appreciation of beauty via sensory perception, sense-making, and organising stimuli in meaningful and pleasing ways. For this work, we situate aesthetics within mathematics learning by connecting aesthetics with pragmatism. Pragmatism is linked with “art” (Cherryholmes, 1999) and aesthetics. Dewey (1934) emphasises the importance of continuity between experiences associated with artistic works and the artistic process(es) arising in everyday events, actions, and challenges. The connection between experience and process gives rise to conceptions, actions, decisions, and imagined consequences that are “desirable, pleasurable, satisfying, and beautiful” (Cherryholmes, 1999, p. 29). Take, for example, people who enjoy the sights and sounds of backhoes excavating dirt, or a mechanic who appreciates the products of their work and feels reverence for the materials and tools they use. Dewey (1934) states that recognising aesthetics as part of everyday living reveals aesthetics as receptive and holistic events, creating a felt sense of satisfaction and/or pleasure in the relationships that comprise wholes and parts, order and integration.

Empirical research on aesthetics in fields such as cognitive, affective psychology, and neuroscience includes studies on beauty, aesthetic pleasure, feeling, and preference (e.g., Brielmann & Pelli, 2018; Brielmann et al., 2017, 2021), as well as perception, evaluation, and the creation of art (e.g., Curry, 2023; Ho et al., 2014; Jacobs & Hustmyer, 1974; Jagodzinski & Wallin, 2013; Wood, 2019). We take a more basic or radical approach than these to aesthetics. Our research seeks to recognize, from the Latin recognoscere, “to recall to mind, to know again”, by returning to the root meanings of aesthetics and exploring them within school mathematics towards enabling deaf-centric pedagogies for young children’s spatial development conducive to STEAM education.

Spatial cognition, its development and reliance on images as fully embodied events, comprises a multiplicity of interrelated ideas, conceptions, and tools (National Research Council, 2006, 2009; Newcombe et al., 2013) critical to mathematics and mathematical understanding (Smith, 1964), especially in the early years (e.g., Clements & Sarama, 2011; Davis & SRSG, 2015). Mathematics can be understood as a “special kind of language through which we communicate ideas that are essentially spatial” (Clements & Sarama, 2011, p. 134). Moreover, because geometric concepts underlie mathematics, mathematical understanding necessitates working spatially with images (e.g., Freudenthal, 1973; Mandelbrot, 1982; Tahta, 1989). Mathematics, as a school subject, and geometry as a distinct area within mathematics specifically concerned with properties of space and spatial relations, hold tremendous potential for teaching and developing children’s spatial thinking in depth (Larkin et al., 2016; Lesh, 1976; National Council of Teachers of Mathematics, 2006; Tahta, 1980, 1989).

Yet there remains a need for theoretical and empirical studies to inform how educators and teachers might conceptualise, observe, and assess deaf1 learners’ spatial-mathematical understandings within geometry, especially how the body contributes to this area of their cognitive development (Healy, 2015; Lang & Pagliaro, 2007; Thom & Hallenbeck, 2021, 2022). We wonder what further insights concerning spatial-mathematical processes, skills, and tools can extend what we know about deaf children’s aesthetics, that is, their ways of perceiving, sensing, and learning to “mak[e] sense of sensorimotor activity” (Thom & Hallenbeck, 2022, p. 130), concerning the embodiment and growth of their geometric understandings.

2.3. Spatial Cognition as Geometric, Embodied, and Aesthetic

Despite the role that spatial skills play in understanding mathematics (Mix et al., 2016), particularly in the early years (Gilligan et al., 2017; Verdine et al., 2017), and increasing evidence revealing the value of spatial development (Hawes et al., 2017; Lowrie et al., 2017), little attention is paid to this topic in elementary curricula (Davis & SRSG, 2015), especially geometry (McGarvey et al., 2024; Sinclair et al., 2016). Even less emphasis is given to this kind of spatial development concerning deaf students (Edwards et al., 2013; Pagliaro & Ansell, 2002, 2012; Pagliaro & Ansell, 2012; Parasnis et al., 1996; Watts, 1979). Further still, few studies assess deaf students’ spatial actions as they relate to their mathematical understandings (Gottardis et al., 2011; Kelly et al., 2003; Zarfaty et al., 2004), much less the embodiment of them (Healy, 2015; Krause, 2018, 2019; Lang & Pagliaro, 2007; Thom & Hallenbeck, 2021). The aim of this study is to observe and assess a deaf child’s spatial-geometric understanding within mainstream mathematics contexts. In doing so, we wish to move beyond that which can be seen and/or told regarding deaf students’ spatial-geometric thinking (Thom & Hallenbeck, 2021) and towards that which has been intuited as the “essence” or character of geometry (Tahta, 1980; Thom, 2018) via aesthetics within geometry.

To begin, we take embodiment as central to ensuing aesthetics in geometrical thinking and spatial cognition. In contrast to hands-on activities that employ concrete objects or visual representations, mathematician and mathematics educator Tahta (1980) asserts that to think mathematically involves making meaningful images, working with them, and thinking geometrically. Tahta (1980) describes “making mind-pictures… rhythmic curve-stitching… play materials… imaginative drawing and dance… animated films… refuses to atomise geometry into a series of written exercises but seems to offer a seamless web of thought, action and feeling… to… explore [geometry] in its own terms” (emphasis added, p. 7). Important to note is that images are not limited to pictures, words, signs, or even gestures but include contextually grounded thoughts, actions, and feelings. And while we should not assume that all the meanings a child holds for a given mathematical image can ever be known, represented, or externalised, this assumption neither denies the presence of such images nor discounts their contribution to a child’s spatial-geometric thinking (Thom, 2018). Rather, mathematical images as understandings or ideas grounded in and emerging from topics and contexts give rise to webs of thoughts, actions, and feelings (Tahta, 1980); understandings and ideas whether correct, incorrect, or otherwise, open possibilities that can inform, shape, and become children’s embedded and bodied (i.e., embodied) mental, verbal, signed, written, or physical mathematics.

Mathematics educators Pirie and Kieren (1989) contend that all mathematics, knowing and understanding are embodied, complex, and levelled yet nonlinear. The authors underscore the importance of inquiring into the nuances of learners’ mathematical understandings, the forms they take (e.g., physical movement, gesture, drawing, and symbolic text), and how they shape previous and subsequent thinking, including learners’ more advanced levels of thinking, regardless of employing theorems or proofs (e.g., Pirie & Kieren, 1994; Thom & Pirie, 2006). In a more recent study featuring the ways embodiment contributes to spatial-geometric reasonings, the geometries of three young (hearing) learners materialise as continuous “creative process[es] of (re)(con)figuring space” (Thom, 2018, p. 131). It is this notion of (re)(con)figuring space as creative processes that we wish to further examine within the field, both in terms of what Tahta (1980) refers to as “a seamless web of thought, action and feeling” (p. 7) and the need to “explore [geometry] in its own terms” (Tahta, 1980, p. 7). Next, we discuss embodiment as aesthetics, its potential to inform STEAM, and deaf education.

2.4. STEAM Education: Aesthetics, Deaf Aesthetics, and Deaf Education

STEAM education has drawn fire for its incidental incorporation of arts education into STEAM projects (Sanz-Camarero et al., 2023; Graham, 2021). It is suggested that the undervaluation of the arts in STEAM is an epistemological problem, in that artistic knowledge, being deeply subjective, is at variance with the more objective (i.e., more visible) aspects of science, technology, engineering, and mathematics (Sanz-Camarero et al., 2023). Such a view gives rise to a bifurcated notion of reality that reinforces a mind–body dualism.

Alternatively, reinterpreting aesthetics as the “A” in STEAM avoids engagement with dichotomous views that posit reality versus the mind and the mind as contradictory to the body. This study, motivated by our intention to better understand aesthetics associated with deaf children’s spatial cognition within geometry, conceives aesthetics as necessary for embodied learning and as that which emerges and evolves phenomenologically from live(d) experience (Henry, 1999, 2009; Kandinsky, 1926/1979; Merleau-Ponty, 1945; Merleau-Ponty & Lefort, 1968). Within cognitive science, Varela et al. (2016) theorise embodied reflection as a specific kind of mind–body unity, a cultivation of a mindfulness or awareness that implicates live(d) experiences. In these ways, we conceive aesthetics as a form of reflection that attends to both our embeddedness and bodiedness within social–cultural spheres (Varela et al., 2016) as well as the more-than-human world (Abram, 1996, 2010). Understanding aesthetics as arising from action, eliciting emotion, and, in some cases, a sense of fulfilment, promotes aesthetics as undivorced from the mind, body, and everyday life (Dewey, 1934). More radically, aesthetics as immanent occasions knowledge of what the mind and body can do (e.g., Nancy, 2006; Varela et al., 2016) and assumes “sense has to have a body… in order for the body to make sense” (Nancy, 2006, p. 65). This does not mean that it is only the body that senses and makes sense, but for the possibility to make sense and use language, we must have a body. More than and different from “hands-on” activities that incorporate manipulatives designed to model mathematics concepts and operations, it is the material body that interfaces with the (im)material world which makes mathematics and mathematical experience possible.

For this study, we further frame our perspective within the “big three” in deaf education: deaf axiologies (Skyer, 2021), deaf ontologies (Thoutenhoofd, 2000), and deaf epistemologies (Hauser et al., 2010). This frame allows consideration for the ways that deaf aesthetics (Skyer, 2021) can inform STEAM education concerning deaf students and, fundamentally, through mathematics education.

Skyer (2021) identifies the notion of aesthetic immanence in practical ways through deaf aesthetics, which include heightened ocular-centric perception, multimodality, and interactivity within deaf cultural, social, and learning milieus. Deaf aesthetics as grounded in deaf biosociality evolves from deaf bodies interacting with the world. Interactions amongst deaf bodies, other bodies, (im)material objects, semiotic resources, and languages are shaped first by deaf axiological commitments. Deaf people value seeing over hearing (ocularcentrism), place seeing as a means to organise linguistic structures in sign language (Taub, 2001), which in turn, guide the development of cognition, knowledge, social interactions (Vygotsky, 1997), and engagement in community life. An example of a deaf axiological commitment lies in ensuring the understanding of others as an ethical orientation due to the realities that a deaf child may have limited opportunities to encounter a word, a concept, or a schema through hearing. Ensuring understanding among the deaf is a value repeatedly demonstrated in gatherings of deaf people at conferences, sporting events, meetings, and deaf-centric events (Green, 2015; Friedner, 2016). In a world largely dominated by audiocentrism but increasingly becoming more spatial, deaf axiology includes commitment to the values associated with seeing as sensed, cohering with Tahta’s characterisation of images as “a seamless web of thoughts, actions, and feelings” (Tahta, 1980, p. 7). Being able to apprehend the world primarily through ocularcentrism is a deaf ontological position (Thoutenhoofd, 2000). Finally, a deaf epistemology that flows from deaf axiology and deaf ontology involves amplified ocularcentrism, multimodality (Kurz et al., 2021; Golos et al., 2024), and social interactivity (Friedner & Kusters, 2015).

While deaf aesthetics begin with the deaf body as it moves, thinks, feels, and interacts with the world, there are implications for STEAM education, especially spatial-mathematical cognition. Spatial cognition as dynamic processes, skills, and tools (Newcombe et al., 2013) shape how we know, think, and interact with the world including mathematics. More simply, spatial cognition entails the ways we imagine, make sense of, and manipulate shapes, objects, and events (Uttal & Cohen, 2012; Uttal et al., 2013). Deaf aesthetics thus lends itself to aesthetics within geometry as core to spatial cognition within STEAM education. We wonder what possibilities emerge for observing young deaf children’s perceiving, sensing, and making sense with their eyes, mind’s eye, multimodalities, and interactivity when geometry is taken up in aesthetically engaged and embodied manners. Reflected in the question is mathematician and teacher Paul Lockhart’s view that mathematics is all about crafting beautiful ideas. Lockhart (2009) contends that too often K-12 learners are denied opportunities to engage with mathematics in ways that necessitate questioning, exploring, and getting to know what “mathematics looks like and feels like” (emphasis added, p. 26), and even less frequent are experiences whereby knowledge and skills result from “the thrill, the joy, even the pain and frustration of the creative act” (Lockhart, 2009, emphasis added, p. 28).

So, while spatial cognition is inherently mathematical and emphasises core spatial processes, skills, and tools (Newcombe et al., 2013) across all STEAM disciplines, this research also emphasises aesthetics, inherent in the arts, as a more prime or basic “A” within STEAM, essential to spatial skills development. Moving deeper into the “A” of STEAM education, beyond common applications of the visual arts such as drawing or model making, we ask how deaf aesthetics might contribute to further understanding spatial cognition as a seamless web of geometric thinking, acting, and feeling (Tahta, 1980).

3. Theoretical–Methodological Framework

This study explores the aesthetic ways Evan’s spatial activity, as embodied cognition, arises and evolves as he works on geometric tasks. Integral to our theoretical–methodological framework are the meanings for aesthetics (i.e., “pertaining to sensory perception” and “perceive, sense, learn”); aesthetics in mathematics (Lockhart, 2009; Tahta, 1980); aesthetics as the philosophy of art (Curry, 2023; Henry, 2009; Kandinsky, 1926/1979); aesthetics as phenomenological experience (Henry, 1999, 2009; Jagodzinski & Wallin, 2013; Merleau-Ponty, 1945; Merleau-Ponty & Lefort, 1968; Nancy, 2006); and deaf aesthetics (Skyer, 2021, 2023). This case study tracks and maps the geometric understandings that facilitate and are embodied in Evan’s spatial thinking. Since aesthetics involve embodied learning (Decoursey, 2018; Gulliksen, 2017; Light, 2024; Maivorsdotter & Lundvall, 2009; Odendahl, 2021; Todd et al., 2021; Xu et al., 2025), specific attention is given to the instances during which Evan enacts geometric understandings and how these instances instantiate aesthetic experiences as he works on the geometric tasks.

The Pirie–Kieren Dynamical Theory and Model for the Growth of Mathematical Understanding

This work as an initial study, which examines how aesthetics shape a deaf child’s spatial-geometric activity, utilises the Pirie–Kieren Dynamical Theory and Model for the Growth of Mathematical Understanding (Pirie & Kieren, 1989, 1992, 1994). Theoretically and methodologically coherent with the design of this research, Pirie and Kieren conceive mathematical understanding as an embodied fluid, back-and-forth, levelled yet nonlinear, and recursive process (Davis et al., 2025a; Irvine, 2023; Kieren, 1990; Pirie & Kieren, 1989, 1994). Importantly, Pirie and Kieren describe their theory as “a theory for growth of understanding of a specified topic by a specific person or group. Thus, the Model serves as a theoretical lens through which an individual’s or group’s mathematical understanding can be illuminated, described, and assessed as it unfolds … in the forms in which it exists” (Thom & Pirie, 2006, p. 186), moment to moment. Further, observation and analysis begin where a learner or group happens to be at a given point in time and which includes everything they bring to the task at hand (Pirie & Kieren, 1994). Pirie and Kieren identify this place of knowing as Primitive Knowing.

The Model (Figure 1) and Theory feature eight nested levels. Moving outwards from the innermost realm, Primitive Knowing is “the starting place for the growth of any particular mathematical understanding” (p. 170). Next is Image Making. Image Making allows one to “make distinctions in previous knowing and use it in new ways” (Pirie & Kieren, 1994, p. 170). Image Having lies on the other side of a Don’t Need Boundary (as indicated by a thick black line) and is when a learner or learners “use a mental construct about a topic without having to do the particular activities which brought it about” (Pirie & Kieren, 1994, p. 170). The Don’t Need Boundaries demarcate when a learner or group of learners “does not need to rely on the more specific, inner understandings that gave rise to the outer knowing.” (Thom & Pirie, 2006, p. 190). Property Noticing serves to “manipulate or combine aspects of images to construct context specific, relevant properties” (Pirie & Kieren, 1994, p. 170). Formalising is described by Pirie and Kieren (1994) as understanding which lies beyond another Don’t Need Boundary and is distinguished as activity that “abstracts a method or common quality from the previous image dependent know how which characterized noticed properties” (p. 170). Observing involves “reflect[ing] on and coordinat[ing] formal activity and express[ing] coordinations as theorems” (Pirie & Kieren, 1994, p. 171). Beyond yet another Don’t Need Boundary is Structuring, understanding that entails “formal observations as a theory” (Pirie & Kieren, 1994, p. 171). At the outermost level, Inventising characterises a learner or group of learners “break[ing] away from preconceptions … and creat[ing] new questions [that] might grow into a totally new concept” (Pirie & Kieren, 1994, p. 171).

Figure 1. Pirie–Kieren Model of a Dynamical Theory for the Growth of Mathematical Understanding.

Figure 1. Pirie–Kieren Model of a Dynamical Theory for the Growth of Mathematical Understanding.

As nested, each layer includes all inner levels, yet each layer is also constrained by the outer levels. Within the Pirie–Kieren Model, depth and breadth of mathematical growth develops as understanding moves outwards and inwards. Outward movement enables possibilities for more generalised mathematical meaning to emerge. Inward movement makes for the return to previous levels of understanding. Pirie and Kieren identify such returns as “Folding Back.” Folding Back, as vital to growth, includes retrieving mathematical experiences or information and accounts for the ways a learner or group of learners “thicken” their understanding to later work at outer layers through reconstructing, reintegrating, or re-evaluating known mathematics (Martin & Pirie, 1998; Pirie & Kieren, 1991).

The Theory/Model emphasises the growth of understanding of a specified topic by a specific person or group, providing a conceptual lens through which to observe, describe, assess, and evaluate embodied spatial-geometric activity as it emerges, unfolds, and materialises (e.g., multimodally as gestures, language, models, diagrams, and text) within mathematical settings. For this study, we examine how the Theory/Model can be used in deaf education research to illuminate the mathematical understandings embedded and aesthetically bodied as Evan’s spatial activity concerning 3D objects and 2D–3D rectangular arrays.

4. Watching Evan’s Spatial-Geometric Understandings Grow

4.1. Methods

The observation and analysis for this qualitative study involved working directly with the materials (e.g., everyday objects, Euclidean blocks), generated artefacts (e.g., cube models) and video recording Evan as he worked on the geometry tasks during a whole class lesson, with a partner, and in a small group setting. First, we analysed the data against current spatial reasoning literature to identify and code Evan’s spatial actions as he engaged in the three geometric tasks: (De)constructing (de/re/composing; re/arranging; and sectioning); Moving (rotating), Situating (dimension shifting; locating, and orienting); Sensating (visualising, tactilizing); and Interpreting (modelling, symmetrizing, comparing, and relating) (Davis & SRSG, 2015). We then considered Evan’s spatial actions in relation to the Pirie–Kieren Theory, observing, documenting, and assessing his geometric understandings as they emerged and evolved within and across the eight Pirie–Kieren Model levels (Pirie & Kieren, 1994), including instances wherein Folding Back occurred.

We examined how Evan develops and integrates informal and more formal mathematical meanings as he interfaces with (im)material mathematics objects2 using multiple modes of access such as sensing, perceiving, and intuiting while he thinks, acts, and feels (Tahta, 1980).

4.2. Data Sources

4.2.1. Study Context

This study features excerpts from two mathematics lessons during Evan’s Kindergarten class and one lesson during Grade 2 at a public elementary school in Western Canada. The children in these classes come from diverse socio-cultural, linguistic, and economic backgrounds. The First Author, research assistants, teachers (classroom and TOD), educational assistants, and ASL interpreter collaborated in designing the lessons as well as facilitating Evan and his peers’ work while they engaged in Tasks 1–3, which were each approximately 45 min.

4.2.2. Participants

Evan has mild to moderate hearing loss. One of his parents is deaf. Evan wears two hearing aids and uses a DM system. He mostly communicates through spoken/print English and uses some Sign-Supported Speech. According to his teachers, Evan demonstrates delays in both receptive and expressive language and is “approaching [provincial] curricular expectations in mathematics”. Evan’s Kindergarten/Grade 2 classes included deaf (N = 2, Kindergarten; N = 2, Grade 2) and hearing peers (N = 19, Kindergarten; N = 20, Grade 2). According to the children’s teachers, student competencies ranged from developing to fully meeting the Kindergarten/Grade 2 provincial curricular expectations. Although Evan has mild-to-moderate hearing loss and communicates primarily via spoken English, we position him within the broader framework of deaf education research due to his experiences and delayed expressive/receptive language development.

Evan and his classes were selected (Enns, 2017) in consultation with the school TOD and district itinerant TOD and he was invited to participate. As deafness is a low-incidence disability, conducting mainstream in situ classroom-based research is additionally constrained by securing informed parental and teacher consent. Given that Evan and his parents, his Kindergarten and Grade 2 classroom teachers, the school and district TOD, ASL interpreter, and Evan’s peers and their parents supported these studies, we took both opportunities to carry out our research. Further, while the first study takes place when Evan is in Kindergarten and the second study occurs when he is in Grade 2, the design of the tasks and theoretical–methodological framework employed accommodate the cross sectional approach of this research.

4.2.3. Data and Data Collection

Tasks 1, 2, and 3

The design of the tasks was informed by early years spatial reasoning research (e.g., Clements & Sarama, 2014; Davis & SRSG, 2015). Task 1—Where Does It Belong? was presented as a whole class activity. Drawing a square pyramid from a bag of everyday (e.g., onion) and Euclidean (e.g., cone) objects, Evan was to consider, decide, and provide at least one reason for why the pyramid should either be included in an existing group with other objects or in a new group. Task 2—Down by the Station, taking approximately 30 min, required Evan and classmate Alaina (deaf) to build a “train”. Placing blocks as “train cars” one behind another, each car had to be different in at least one way from the previous car. For this session, Evan and Alaina worked with their TOD Mrs. Johnson in her room across the hall from the classroom. Task 3—Pack Them Up, taking approximately 45 min, occurred two years after the first and second tasks. Here, the First Author and ASL interpreter worked with Evan and two students (one deaf, one hearing) in a room with three other smaller groups with 3–4 students each, now in Grade 2, to find ways 12 chocolates could be arranged in a rectangular package.

5. Results and Analysis: Towards Understanding Evan’s Understandings

This section highlights key moments in Evan’s work from the three lessons. We share these findings in three ways: as descriptive vignettes, summarised in Table 1, Table 2 and Table 3, and mapped onto the Pirie–Kieren Model in Figure 2, Figure 3 and Figure 4. Table 1, Table 2 and Table 3 and Figure 2, Figure 3 and Figure 4 illustrate our analyses (Pirie, 1996) of Evan’s activity, spatial actions (Davis & SRSG, 2015), and geometric understandings as observed, assessed, and evaluated within each of the tasks and across all three tasks using the Pirie–Kieren Theory/Model (Pirie & Kieren, 1994). Here, we offer in-depth accounts that attempt to illuminate the dynamical aesthetics of Evan’s spatial-geometric work and present his understandings as descriptive vignettes, multi-layered transcriptions, and maps utilising the Pirie–Kieren Model.

5.1. Task 1: Where Does It Belong?

Vignette 1, Sequence 1: Evan looks at the square pyramid as he decides whether it belongs in an existing group (i.e., category) with other objects or in its own group. Evan turns the pyramid onto its side and then rotates it in his hands. Noreen, sitting beside Evan, turns to look at Evan and, at the same time, places her finger on the apex of a cone. Evan looks at the cone and the pyramid (the class had identified this pyramid as different from the objects sorted thus far). Evan looks at the cone and the pyramid.

Sequence 2: The pyramid falls from Evan’s hands. He tries to roll it along the floor. Picking the object up, Evan holds it and looks down at the square face (i.e., bottom of the pyramid) which is now visible, shifting his gaze back and forth from the pyramid to the cone. Noreen, who is holding the cone, continues watching Evan work, then removes her finger from the apex of the cone.

Sequence 3: Evan continues rotating the inverted pyramid in his hands. Noreen looks at the cone and places her fingers back on the apex. Still facing Noreen, Evan looks at the pyramid base, then to the rest of the sorted objects (e.g., cylinder, triangular prism), including the cone in Noreen’s hands. He tells the class that the square pyramid belongs “in its own group.” Evan’s teacher hands him a sheet of paper onto which he places the pyramid, signifying it as distinct from the other objects.

Sequence 4: When asked why the pyramid is different from the other objects, Evan articulates one similarity and three differences that distinguish the pyramid from the cone. While holding the pyramid, Evan takes the cone from Noreen and inverts both objects so the two bases are visible. He presses the pyramid base against the base of the cone, demonstrating both surfaces as flat. Evan then explains that the base of the cone is “ova[l] (sic)” whereas the pyramid’s base “isn’t”. Next, Evan points with his finger, touching each of “these” four vertices on the pyramid base and exclaiming, “Ouch! I feel pointy, sharp.” Finally, Evan traces around the bases of the objects, revealing the circumference of the cone and the rectangular perimeter of the pyramid.

Table 1. This table summarises Evan’s spatial-geometric activity for Task 1 as observed, identified as spatial actions (Davis & SRSG, 2015), and assessed according to the Pirie–Kieren Theory (Pirie & Kieren, 1994).

Sequence 1

Spatial Actions

Moving as rotating the pyramid. Sensating as tactilizing and visualising. Evan looks at the pyramid as he turns it in his hands while also looking at the cone.

Corresponding Pirie–Kieren Level

Image Making of the pyramid and the cone.

Sequence 2

Spatial Actions

Moving as rotating the pyramid. Sensating as tactilizing and visualising. Evan looks at the pyramid as it falls while also looking at the cone. He attempts to roll the pyramid. Picking the pyramid up, he inverts it and looks at the base. Interpreting as comparing (and possibly relating) the pyramid with the cone (e.g., apex).

Corresponding Pirie–Kieren Level

Image Making of the pyramid and cone in terms of whether each object rolls, does not roll, and the pyramid’s base. Plausible is the emergence of Image Having regarding these attributes and Property Noticing as comparing/relating them with each other.

Sequence 3

Spatial Actions

Moving as rotating the pyramid. Sensating as tactilizing and visualising when Evan looks and turns the pyramid in his hands. Interpreting as comparing when Evan looks at the pyramid’s base, then to the rest of the sorted/classified objects, and says the pyramid belongs in “its own group”.

Corresponding Pirie–Kieren Level

Folding Back to Image Making of the pyramid’s base then to Property Noticing of the pyramid in relation to the other sorted/classified objects.

Sequence 4

Evan turns the pyramid over, takes the cone from Noreen, inverts it, and presses the bases together. He explains, “because…this one is ova[l] and this one isn’t”. Evan then points to the corners of the bottom of the pyramid as he says, “These. Ouch! I feel pointy, sharp”, tracing around the edges of the objects’ bases with his finger.

Spatial Actions

Moving as rotating the pyramid and cone. Situating as dimension shifting, locating, orienting the pyramid and cone, and focusing on the bases of the pyramid and cone. Sensating as tactilizing and visualising, touching, and looking as he compares the cone with the pyramid. Interpreting as comparing and possibly relating the cone with the pyramid (e.g., flat surfaces).

Corresponding Pirie–Kieren Level

Folding Back to Image Having to recollect (e.g., 2D faces for bases of pyramid and cone). Property Noticing as making distinctions about the overall shape of the bases for the pyramid and cone. Property Noticing as distinguishing specific attributes of the cone and pyramid such as the shape of their perimeters and presence/absence of corners.

Table 1. This table summarises Evan’s spatial-geometric activity for Task 1 as observed, identified as spatial actions (Davis & SRSG, 2015), and assessed according to the Pirie–Kieren Theory (Pirie & Kieren, 1994).

Sequence 1

Spatial Actions

Moving as rotating the pyramid. Sensating as tactilizing and visualising. Evan looks at the pyramid as he turns it in his hands while also looking at the cone.

Corresponding Pirie–Kieren Level

Image Making of the pyramid and the cone.

Sequence 2

Spatial Actions

Moving as rotating the pyramid. Sensating as tactilizing and visualising. Evan looks at the pyramid as it falls while also looking at the cone. He attempts to roll the pyramid. Picking the pyramid up, he inverts it and looks at the base. Interpreting as comparing (and possibly relating) the pyramid with the cone (e.g., apex).

Corresponding Pirie–Kieren Level

Image Making of the pyramid and cone in terms of whether each object rolls, does not roll, and the pyramid’s base. Plausible is the emergence of Image Having regarding these attributes and Property Noticing as comparing/relating them with each other.

Sequence 3

Spatial Actions

Moving as rotating the pyramid. Sensating as tactilizing and visualising when Evan looks and turns the pyramid in his hands. Interpreting as comparing when Evan looks at the pyramid’s base, then to the rest of the sorted/classified objects, and says the pyramid belongs in “its own group”.

Corresponding Pirie–Kieren Level

Folding Back to Image Making of the pyramid’s base then to Property Noticing of the pyramid in relation to the other sorted/classified objects.

Sequence 4

Evan turns the pyramid over, takes the cone from Noreen, inverts it, and presses the bases together. He explains, “because…this one is ova[l] and this one isn’t”. Evan then points to the corners of the bottom of the pyramid as he says, “These. Ouch! I feel pointy, sharp”, tracing around the edges of the objects’ bases with his finger.

Spatial Actions

Moving as rotating the pyramid and cone. Situating as dimension shifting, locating, orienting the pyramid and cone, and focusing on the bases of the pyramid and cone. Sensating as tactilizing and visualising, touching, and looking as he compares the cone with the pyramid. Interpreting as comparing and possibly relating the cone with the pyramid (e.g., flat surfaces).

Corresponding Pirie–Kieren Level

Folding Back to Image Having to recollect (e.g., 2D faces for bases of pyramid and cone). Property Noticing as making distinctions about the overall shape of the bases for the pyramid and cone. Property Noticing as distinguishing specific attributes of the cone and pyramid such as the shape of their perimeters and presence/absence of corners.

Figure 2. Evan’s growth of understanding as observed and assessed during Sequences 1–4 (i.e., S1, S2, S3, S4) for Task 1 according to the Pirie–Kieren Theory and as mapped on the Model.

Figure 2. Evan’s growth of understanding as observed and assessed during Sequences 1–4 (i.e., S1, S2, S3, S4) for Task 1 according to the Pirie–Kieren Theory and as mapped on the Model.

5.2. Task 2: Down by the Station

Vignette 2, Sequence 1: Evan and Alaina sit in front of their built train. Behind the steam engine is a line of nine train cars. Mrs. Johnson (TOD) asks Evan what is different about the second (cube) train car and the first (sphere) train car. Evan responds, “I don’t know.”

Sequence 2: Mrs. Johnson places the sphere in Evan’s left hand and the cube in his right hand. Evan looks at the blocks. Putting down the sphere, Evan touches the four corners of one of the cube’s square faces with his hands. He then announces “edges (sic)” as he holds and looks at the cube, this time placing his fingertip on each of the vertices.

Sequence 3: With the cube still in his hand, Evan points to one of the faces on the side of the block and says, “faces”. Next, straightening and flattening his index and second finger, he sweeps them across the square surface.

Sequence 4: Evan cups his hands and wraps them around the cube. Gazing down at the block, he presses his fingertips against the vertical edge that is underneath them. Looking at his hands wrapped around the block, Evan says “edges”. As he says “edges”, he watches his fingertips glide along the edge, and before they reach the end, his gaze shifts from one directed downwards to straight ahead. Attending to the edge some more, Evan repeats the downward motion with his index finger, no longer curved, now straight.

Table 2. This table summarises Evan’s spatial-geometric activity for Task 2 as observed, identified as spatial actions (Davis & SRSG, 2015), and assessed according to the Pirie–Kieren Theory and Model (Pirie & Kieren, 1994).

Sequence 1

When asked what is different about the second train car (i.e., cube) from the first car (i.e., sphere), Evan says “I don’t know.”

Spatial Actions

No action.

Corresponding Pirie–Kieren Level

Evan’s response “I don’t know” is his starting place or Primitive Knowing.

Sequence 2

TOD places the sphere and cube in each of Evan’s hands. Evan looks at them and turns them about. Putting the sphere down, he gazes at the cube. Evan announces “edges [sic] [corners]” as he places his fingertip on the four corners.

Spatial Actions

Deconstructing as decomposing the cube (e.g., corners). Situating as dimension shifting by locating and verbally identifying the four corners or vertices of the cube. Moving as rotating the cube. Sensating as visualising and tactilizing the cube with his eyes and hands. Interpreting as comparing and relating the presence of vertices for the cube and the absence of them for the sphere.

Corresponding Pirie–Kieren Level

Continuous Image Making of what Evan later demonstrates as Image Having (i.e., “edges [sic] [corners/vertices]”). Property Noticing as the four vertices Evan identifies with his fingertip as points. Folding Back to Image Making to develop what he knows about the cube. Vertices become an Image he then “has” and uses to distinguish the cube from the sphere (Property Noticing).

Sequence 3

Evan points to the side of the block, then grasps the front and says, “faces”; flattening his two fingers, he sweeps them across the surface.

Spatial Actions

Deconstructing as decomposing the cube from object to faces. Situating as dimension shifting from 3D object to 2D faces. Situating as verbally locating the 2D faces of the cube. Sensating as visualising and tactilizing using eyes and hands. Interpreting as comparing and relating when Evan distinguishes the flat surfaces of the cube as different from the sphere.

Corresponding Pirie–Kieren Level

Image Making of the cube’s faces as flat surfaces. Image Having of the cube’s faces, which he uses to Property Notice as a second attribute which distinguishes the cube from the sphere.

Sequence 4

Gazing downward, Evan places his fingertips along one of the edges of the cube. As he says “edges”, he looks at his fingers gliding along the edge. He then stares straight ahead. Looking at the edge of the cube again, he repeats the straight downward motion, this time with his index finger.

Spatial Actions

Deconstructing as decomposing and Situating as dimension shifting the cube from 3D object to 1D lines through locating the cube’s edges. Evan also makes these evident by verbalising “edges”. Sensating as visualising and tactilizing with his eyes and hands, which suggest visualising the movement as seen and felt. Interpreting as comparing and relating, distinguishing the flat surfaces of the cube.

Corresponding Pirie–Kieren Level

Folds Back to Image Making of the cube’s straight edges. As Image Having, Evan uses “edges” to Property Notice, demonstrating them as a third attribute which distinguishes the cube from the sphere.

Table 2. This table summarises Evan’s spatial-geometric activity for Task 2 as observed, identified as spatial actions (Davis & SRSG, 2015), and assessed according to the Pirie–Kieren Theory and Model (Pirie & Kieren, 1994).

Sequence 1

When asked what is different about the second train car (i.e., cube) from the first car (i.e., sphere), Evan says “I don’t know.”

Spatial Actions

No action.

Corresponding Pirie–Kieren Level

Evan’s response “I don’t know” is his starting place or Primitive Knowing.

Sequence 2

TOD places the sphere and cube in each of Evan’s hands. Evan looks at them and turns them about. Putting the sphere down, he gazes at the cube. Evan announces “edges [sic] [corners]” as he places his fingertip on the four corners.

Spatial Actions

Deconstructing as decomposing the cube (e.g., corners). Situating as dimension shifting by locating and verbally identifying the four corners or vertices of the cube. Moving as rotating the cube. Sensating as visualising and tactilizing the cube with his eyes and hands. Interpreting as comparing and relating the presence of vertices for the cube and the absence of them for the sphere.

Corresponding Pirie–Kieren Level

Continuous Image Making of what Evan later demonstrates as Image Having (i.e., “edges [sic] [corners/vertices]”). Property Noticing as the four vertices Evan identifies with his fingertip as points. Folding Back to Image Making to develop what he knows about the cube. Vertices become an Image he then “has” and uses to distinguish the cube from the sphere (Property Noticing).

Sequence 3

Evan points to the side of the block, then grasps the front and says, “faces”; flattening his two fingers, he sweeps them across the surface.

Spatial Actions

Deconstructing as decomposing the cube from object to faces. Situating as dimension shifting from 3D object to 2D faces. Situating as verbally locating the 2D faces of the cube. Sensating as visualising and tactilizing using eyes and hands. Interpreting as comparing and relating when Evan distinguishes the flat surfaces of the cube as different from the sphere.

Corresponding Pirie–Kieren Level

Image Making of the cube’s faces as flat surfaces. Image Having of the cube’s faces, which he uses to Property Notice as a second attribute which distinguishes the cube from the sphere.

Sequence 4

Gazing downward, Evan places his fingertips along one of the edges of the cube. As he says “edges”, he looks at his fingers gliding along the edge. He then stares straight ahead. Looking at the edge of the cube again, he repeats the straight downward motion, this time with his index finger.

Spatial Actions

Deconstructing as decomposing and Situating as dimension shifting the cube from 3D object to 1D lines through locating the cube’s edges. Evan also makes these evident by verbalising “edges”. Sensating as visualising and tactilizing with his eyes and hands, which suggest visualising the movement as seen and felt. Interpreting as comparing and relating, distinguishing the flat surfaces of the cube.

Corresponding Pirie–Kieren Level

Folds Back to Image Making of the cube’s straight edges. As Image Having, Evan uses “edges” to Property Notice, demonstrating them as a third attribute which distinguishes the cube from the sphere.

Figure 3. Evan’s growth of understanding as observed and assessed during Sequences 1–4 for Task 2 according to the Pirie–Kieren Theory and as mapped on the Model.

Figure 3. Evan’s growth of understanding as observed and assessed during Sequences 1–4 for Task 2 according to the Pirie–Kieren Theory and as mapped on the Model.

5.3. Task 3: Pack Them up

Vignette 3, Sequence 1: Two years later, Evan, now in Grade 2, works to find ways that 12 “chocolates” (cubes) can be packaged. Right away, he builds a rod of six cubes, to which he attaches a second rod of six cubes, and creates a 2 × 6 array.

Sequence 2: Evan then connects a rod of four cubes. He uses the rod and measures the length of a model constructed by another group member, which the First Author holds for him: a 2 × 3 array layered onto another 2 × 3 array, effectively a 2 (2 × 3) array. Evan continues building, adding two more cubes to the rod that forms a right angle four cubes long and three cubes wide. He finishes by attaching six more cubes to produce a 3 × 4 array.

Sequence 3: With 12 more cubes in front of Evan, he looks back and forth at the loose cubes and built models [i.e., 1 × 12, 2 × 6, 3 × 4, 2 (2 × 3)]. He proceeds to take apart the 2 × 6 array into two rods of six cubes each. Evan then disassembles one of the 1 × 6 rods and uses four cubes to create a 2 × 2 array. Evan builds another 2 × 2 array by taking apart the second 1 × 6 rod and attaching it on top of the first array. Evan holds up a cube and proceeds to look back and forth at the cube and his second (2 × 2) model. He then connects the cube to the end of the model and the three remaining cubes to form a third 2 × 2 array. Complete, Evan’s model features a 2 × 2 × 3 array, as placed on the table, that sits 2 cubes wide, 2 cubes high, and 3 cubes long.

Table 3. This table summarises Evan’s spatial-geometric activity for Task 3 as observed, identified as spatial actions (Davis & SRSG, 2015), and assessed according to the Pirie–Kieren Theory and Model (Pirie & Kieren, 1994).

Sequence 1

Right away, Evan assembles one rod of six cubes. To this he adds a second rod of six cubes to create a 2 × 6 array.

Spatial Actions

Constructing as composing, arranging, and sectioning rods of cubes. Sensating as visualising and tactilizing, using eyes and hands to construct the 2 × 6 cube array. Interpreting as modelling, symmetrizing, and relating the 12 cubes as a pair of rods containing 6 cubes each.

Corresponding Pirie–Kieren Level

Image Having as two rods of six cubes each and twelve cubes altogether.

Sequence 2

Evan builds a rod of four cubes and measures it against the length of another cube model that contains two 2 × 3 arrays. Evan adds two more cubes to the rod at a right angle to make a width of three cubes. He then fills in the rest of the cubes and produces a 3 × 4 array.

Spatial Actions

Constructing by composing, arranging, and sectioning to create a 1 × 4 then a 3 × 4 cube array. Situating by dimension shifting from a 1D 1 × 4 array to a 2D 3 × 4 array.

Corresponding Pirie–Kieren Level

Image Making of the 1 × 4 and 3 × 4 arrays. Image Having of a different array than the other arrays as 4 cubes long, 3 cubes wide, and 12 cubes altogether.

Sequence 3

Evan looks back and forth at the other models and the 12 cubes in front of him. He disassembles one of the cube models, rebuilding it as three 2 × 2 arrays, and connects them as a 2 × 2 × 3 array.

Spatial Actions

Deconstructing a different cube array by decomposing, recomposing, and rearranging the cubes as three 2 × 2 cross sections of the 12-cube array. Situating as dimension shifting from two 2D arrays (i.e., 2 × 6, 3 × 4) to a 3D array (i.e., 2 × 2 × 3). Interpreting by modelling 12 cubes as a 2 × 2 × 3 array or 3 (2 × 2).

Corresponding Pirie–Kieren Level

Image Making of another array, distinct from the other two models. Image Having of a twelve-cube 3D array that is two cubes wide, two cubes high, and three cubes long.

Table 3. This table summarises Evan’s spatial-geometric activity for Task 3 as observed, identified as spatial actions (Davis & SRSG, 2015), and assessed according to the Pirie–Kieren Theory and Model (Pirie & Kieren, 1994).

Sequence 1

Right away, Evan assembles one rod of six cubes. To this he adds a second rod of six cubes to create a 2 × 6 array.

Spatial Actions

Constructing as composing, arranging, and sectioning rods of cubes. Sensating as visualising and tactilizing, using eyes and hands to construct the 2 × 6 cube array. Interpreting as modelling, symmetrizing, and relating the 12 cubes as a pair of rods containing 6 cubes each.

Corresponding Pirie–Kieren Level

Image Having as two rods of six cubes each and twelve cubes altogether.

Sequence 2

Evan builds a rod of four cubes and measures it against the length of another cube model that contains two 2 × 3 arrays. Evan adds two more cubes to the rod at a right angle to make a width of three cubes. He then fills in the rest of the cubes and produces a 3 × 4 array.

Spatial Actions

Constructing by composing, arranging, and sectioning to create a 1 × 4 then a 3 × 4 cube array. Situating by dimension shifting from a 1D 1 × 4 array to a 2D 3 × 4 array.

Corresponding Pirie–Kieren Level

Image Making of the 1 × 4 and 3 × 4 arrays. Image Having of a different array than the other arrays as 4 cubes long, 3 cubes wide, and 12 cubes altogether.

Sequence 3

Evan looks back and forth at the other models and the 12 cubes in front of him. He disassembles one of the cube models, rebuilding it as three 2 × 2 arrays, and connects them as a 2 × 2 × 3 array.

Spatial Actions

Deconstructing a different cube array by decomposing, recomposing, and rearranging the cubes as three 2 × 2 cross sections of the 12-cube array. Situating as dimension shifting from two 2D arrays (i.e., 2 × 6, 3 × 4) to a 3D array (i.e., 2 × 2 × 3). Interpreting by modelling 12 cubes as a 2 × 2 × 3 array or 3 (2 × 2).

Corresponding Pirie–Kieren Level

Image Making of another array, distinct from the other two models. Image Having of a twelve-cube 3D array that is two cubes wide, two cubes high, and three cubes long.

Figure 4. Evan’s growth of understanding as observed and assessed during Sequences 1–3 for Task 3 according to the Pirie–Kieren Theory and as mapped on the Model.

Figure 4. Evan’s growth of understanding as observed and assessed during Sequences 1–3 for Task 3 according to the Pirie–Kieren Theory and as mapped on the Model.

6. Discussion

Our observations, analyses, assessments, and evaluations highlight Evan’s spatial activity as interrelated with his geometric understandings, emergently grounded in and aesthetically arising as he makes sense of what he senses and perceives. We see this study contributing to laying the groundwork for more in-depth research that responds to deaf aesthetics and the growth of deaf students’ spatial-geometric understandings.

Evan’s work across Tasks 1, 2, and 3 offers up-close instances involving the aesthetics of his spatial-geometric understandings; that is, how his understandings for 0–3D mathematics objects can be observed as rooted in and growing from that which he experientially senses, perceives, and thus learns firsthand. For example, we observe the aesthetic manners in which the pyramid falls from Evan’s hands and lands on the carpet and, later, when he attempts to roll the pyramid on its side after having looked back and forth at it and the cone. Evan’s activity, as observed, cannot be simplified or parsed out as more or less sophisticated spatial-geometric understandings. Instead, the beautiful mathematics ideas and meanings or “images” (Pirie & Kieren, 1989; Tahta, 1980) that Evan actively crafts and articulates for curved and flat planes, circular and rectangular faces, and vertices happen dialectically as part and whole. Illuminated are the ways these instantiations reveal how “abstraction and sensation are inherently coincidental … complementary and interrelated [phenomena] rather than disparate” (Thom, 2018, rearranged, p. 154).

Evan’s mathematics as geometric, wholly sensorial, perceptual, epistemic, informal, formal, concrete, and abstract emphasise the dynamic co-emergent and cohesive nature of his Primitive Knowing, Image Making, Image Having, Property Noticing, crossing a Don’t Need Boundary, and Folding Back (Pirie & Kieren, 1994). For example, Image Making plays different roles in Evan’s spatial-geometric activity. Sometimes Image Making emerges as Evan develops new conceptual meaning (e.g., comparing attributes of the square pyramid in relation to other objects and, particularly, with the cone). While in other instances, Image Making results from Evan’s Folding Back as a furthering or “thickening” of meanings for concepts—images he already “has” (e.g., “faces” that then develop in relation to the cube as flat 2D surfaces). These understandings as inward movements can be seen as giving rise to later meanings that connect with earlier ideas (e.g., “faces” as Image Having). We observe these new and more sophisticated understandings as also facilitating Evan’s formal meaning making and exemplifying the creative back and forth processes in which he (re)crafts ideas. Each and all, dialectically as partial and whole, enable Evan to work fluidly and move seamlessly within as well as across the four Pirie–Kieren levels and a Don’t Need Boundary (e.g., Image Having and Property Noticing, explicating the vertices, faces, and edges of a cube vs. sphere).

Whether sorting and classifying attributes of 3D objects or generating rectangular arrays for 12 cubes, Evan’s Image Making reveals how he

Com[es] to understand… when [he] performs actions — mental or physical — in order to create some “idea” of the new topic. Here, understanding grows from making distinctions through mathematical actions … the intent of working at this level is to give rise to the creation of new mathematical “images,” which may exist in mental, verbal, written, [signed], or physical forms.

(Thom & Pirie, 2006, p. 189)

Moreover, in each of the three tasks as Evan continuously works at a particular level, including when Folding Back to work at an inner level, Evan’s activities serve

Not as just the recollection of a mathematical experience or piece of information, but as providing a means by which [he] reconstruct[s], reintegrate[s], or re-evaluate[s] known mathematics so that [he] may function in the outer layers with a ‘thicker’ understanding.

(Thom & Pirie, 2006, p. 187)

Aesthetically speaking, Image Making can be seen as one aspect of what Lockhart (2002) refers to as the “hard, creative work” (p. 11) that is mathematics and, in this study, one of the contexts in which Evan engages as he geometrically (re)(con)figures space.

We also draw readers’ attention to the fact that, while Evan uses some spoken language, often preceding his verbalizations are emergent, embedded, and bodied movements, gestures, and material models. As embodied understandings, all happen simultaneously as a “seamless web of thought, action and feeling…. [as Evan] …explore[s] [geometry] in its own terms” (Tahta, 1980, emphasis added, p. 7), further reflecting Nancy’s point of immanence and dialectic unity in which “sense has to have a body… in order for the body to make sense” (Nancy, 2006, p. 65). These observations support studies that evidence spatial thinking in mathematics as largely nonverbal, such as dynamic mental and concrete imagery, gestures, models, drawings, and movement (Bussi & Baccaglini-Frank, 2015; Healy, 2015; Radford, 2014; Sinclair & Tabaghi, 2010) and how language (Pulvermüller, 2012) can be understood as “woven in action and originat[ing] from contexts that are sensually experienced and neurologically connected” (Thom, 2018, p. 154).

7. Conclusions: Implications and Considerations for Future STEAM-Related Research in Deaf Education

This study offers an initial exploration into aesthetics, spatial cognition, and ways that the two can inform STEAM areas through deaf aesthetics and spatial-geometric thinking in mathematics. The work extends STEM education research in deaf education (e.g., Kasavan, 2012; Kritzer & Green, 2021; Thom & Hallenbeck, 2021) by considering aesthetics as a more radical “A” within STEAM.

Evan as a deaf student demonstrates how aesthetic ways of perceiving, sensing, and learning enable him to “mak[e] sense of sensorimotor activity” (Thom & Hallenbeck, 2022, p. 130) and give rise to spatial-geometric ideas, processes, skills, and tools (National Research Council, 2006, 2009; Newcombe et al., 2013). Using the Pirie–Kieren Theory and Model (Pirie & Kieren, 1989, 1992, 1994), we demonstrate the potential of the Theory/Model in terms of not only observing, assessing, and evaluating Evan’s spatial-geometric work as nonlinear, recursive, and wholly (em)bodied but, in doing so, recognizing this child’s mathematics, at its core, as inherently aesthetic.

To conclude, we share implications and considerations, occasioned by Evan’s spatial-geometric work, for future deaf education research and classroom mathematics related to STEAM education, research methodologies, and language development.

7.1. STEAM Education Within Deaf Education

Support for arts-integration strategies in education, the relationship between aesthetics as manifested in arts integration or arts education and other subject areas is not well-established or well-understood in STEAM projects (Gettings, 2016). Moreover, the embodied and live(d) experiences of students (and teachers) are understudied areas in educational research (Todd et al., 2021). For this reason, we attended to aesthetics to approach embodied learning rather than arts integration as a pedagogical strategy.

The implications and considerations we present here are intended to integrate the body and the senses within pedagogical practice (Todd et al., 2021). STEAM education is a beginning in this direction and is slowly gaining traction in deaf education. The teacher resource guide, 58-in-Mind (Golos et al., 2024), promotes the theatre in education model (TIE) using interactive drama practices such as acting, role playing, and movement to learn a concept, express understanding, and to reinforce learning and illustrates how this model can be used to teach the parts of the digestive system. The resource provides a lesson plan on Newton’s Laws of Motion using body movement and ASL. The resource also provides an interdisciplinary unit on the movie Hidden Figures, which includes the development of spatial awareness as the deaf students are asked to take photographs of hidden geometric shapes. The science portion of this project requires students to study the seasonal appearances of constellations which may or may not be hidden according to weather patterns.

Rusilowati et al. (2020) developed a STEAM-Deaf learning model in relation to the building of Rube Goldberg machines and found that this activity was highly effective in developing understanding and creativity within the field of science. Mystakidis et al. (2023) explore the delivery of a transdisciplinary STEAM project with deaf high school students using immersive technologies such as virtual reality and the Metaverse (a 3D version of the internet) to construct a virtual exhibit on STEAM topics including the environment and sustainability, diversity, inequality and social justice, science (the human brain), technology, and culture.

For us as hearing and deaf researchers, educators, and teachers, current presentations of STEAM in deaf education prompt new questions. For instance, in deaf education, STEAM principles predominantly use narratives available through theatre in education, role playing, movement, and connecting disparate topics in a virtual exhibit to learn (Golos et al., 2024; Kasavan, 2012; Mystakidis et al., 2023; Rusilowati et al., 2020). However, Evan’s work and the ways he engages in sensing, perceiving, and intuiting as he thinks, acts, and feels (Tahta, 1980) with (im)material mathematics objects create opportunities for mathematics teaching and learning to become “more than” and “different from” using hands-on manipulatives to represent and show-by-telling formal concepts. In what ways does Evan’s work serve as provocation for deaf-centric aesthetics-based mathematics pedagogy?

Furthermore, there are decades of research devoted to explicating conceptual relationships among bodily experiences, metaphors, and narratives (Egan, 2005; Lakoff & Johnson, 1999; Taub, 2001) including within mathematics (English, 2013; Lakoff & Núñez, 2000; Núñez, 2008; Pimm, 1987; Sfard, 1994). New queries that emerge from this study include the following: How might metaphors and narratives contribute to perceiving, sensing, and intuiting, thereby embodying mathematical ideas? In what ways can perception, sensation, (em)bodied learning, and abstraction work together to evolve deaf-centric aesthetics, mathematics, and pedagogies?

Kurz et al. (2021), in their guidelines for educating multilingual students, emphasise the importance of multimodality in addressing the varying abilities and needs of deaf students. Skyer (2021) stresses the increased ocularcentrism, multimodality, and interactivity as embodied learning which leads to the application of deaf aesthetics in multiple disciplines and contexts. Weber (2018) noted that BIPoC deaf youth responded more readily to curricular content such as mathematics (measurement, geometry, and arithmetic operations), social studies, English literature, and deaf studies, expanding their language development and production through arts-based tasks (e.g., theatre, visual arts, stage and set design, and art installations). The emphasis on embodied learning as integral to STEAM deaf education calls for a rethinking of epistemological, ontological, and axiological assumptions concerning deaf people and their learning (Skyer, 2021). Demonstrated in Evan’s activity, we observe possible connections relating multimodality, ocularcentrism, and interactivity through this child’s dynamic and recursive understandings that feature ways of thinking, acting, and sensing as multimodal (kinaesthetic, visual, and spoken/sign language) and ocularcentric. For example, when Evan actively looks at and touches the base of the pyramid and cone in Task 1, or when he interactively builds while geometrically conceptualising 12 as various dimensional rectangular arrays in Task 3. Further still, what are the ways that Evan’s mathematical back-and-forthing as observed, tracked, and mapped onto the Pirie–Kieren Model can be understood as contributing to his interactivity with the different (im)material mathematics objects themselves? How might these ways expand deaf aesthetics and further inform STEAM pedagogy in deaf education?

7.2. Research Methodologies

Evan’s activity, as observed in Tasks 1–3, demonstrates his competency as a mathematics learner characterised as responsive, flexible, and creatively generative. This view contrasts with most studies in the field, which indicate deaf students as less proficient than their hearing peers in mathematics (e.g., Gottardis et al., 2011; Qi & Mitchell, 2012; Traxler, 2000). The findings of this study support extant research towards an alternative view, which asserts deaf students as not only capable of learning mathematics well but as capable of excelling in mathematics when appropriate access, opportunity, and pedagogy are available (Henner et al., 2021; Pagliaro & Thom, 2021). Previous efforts in addressing mathematics assessments, for instance, respond to the impact of accommodations in administering mathematics assessments to level the playing field for deaf children (e.g., Cawthon & Leppo, 2013). As researchers, we took a different approach and investigated manners of teaching, engaging, and assessing that promote the growth of mathematical understanding. This research attends to the pedagogical sensibilities called for when employing the Pirie–Kieren Theory/Model (e.g., see Davis et al., 2025b).

Methodologically, our research as a qualitative case study presents approaches that complement more common quantitative research methods in the field (Henner et al., 2021; Pagliaro & Kritzer, 2013; Pixner et al., 2014; Traxler, 2000). As one of few case studies involving a young deaf child’s spatial-geometric thinking within everyday mainstream contexts, our work contributes knowledge about this topic and possible ways deaf children’s mathematics and mathematical understanding might be conceived and supported beyond the individual accommodations especially in inclusive education contexts. Moreover, we are unaware of other studies in deaf education that implement the Pirie-Kieren Theory/Model. This research as first of its kind, thus innovates empirical, theoretical, and pedagogical potential for the Theory/Model to examine the dynamical nature of diverse deaf students’ mathematical understandings.

7.3. Spatial Cognition and Language Development

Recent cognitive and linguistic studies involving deaf persons suggest spatial cognition as embodied mathematics does not necessarily require advanced language acquisition or a specific language modality (i.e., sign language or spoken language). These findings are indeed compelling (Karadöller et al., 2021, 2022a, 2022b; Sümer & Özyürek, 2020). Spatial cognition as largely nonverbal and thus not solely dependent upon language reinforces the role of embodied learning, deaf aesthetics, and geometric thinking.

This view contrasts with conventional mathematics instruction where the locus predominately fixates on language acquisition as essential for mathematics education, especially in geometry (Sinclair et al., 2016). In practical ways, we see the Pirie–Kieren Theory/Model as also occasioning new opportunities for encouraging deaf aesthetics, and broadening the scope of this study, future research, teaching towards instructional modes, task designs, and the assessment of deaf children’s spatial-geometric understandings predicated upon deaf aesthetics. In doing so, we may optimise spatializing STEAM via geometry, embodied learning, multimodality, ocularcentrism, and interactivity (as facets of deaf aesthetics) while enriching mathematics teaching and learning. Weber (2024) notes that, in her study of deaf youth in their preparation of props for their production of Apple Time, a theatre play, their work included multiple mini-STEAM projects which promoted distributed cognition and language usage in the group. Distributed cognition refers to language as a practice that is distributed across individuals, the material world, tools, and physical spaces, all of which influence and are influenced by our cognition and language practices (Pennycook, 2018). Because the props ranged from puppet fabrication, the construction of puppet screens, developing movements with blankets, masks, necklaces, feathers and headdresses, the students engaged designing, discussing, problem solving, and assisting each other as they worked with fabric, wood, paper, and small electronics and retrofitted large pieces of furniture. All these actions involved the development of spatial thinking, acting, and feeling. Implementation of the Pirie–Kieren Theory/Model could facilitate understanding of the ways in which language and associated cognition evolve as embodied, spatial, and deaf-aesthetic learning.

Emphasising spatial-geometric development can facilitate greater access to mathematics while accommodating a diverse population of deaf students with varying abilities and needs. In turn, this emphasis intends to occasion deaf children’s flexibility and sophistication in understanding mathematics. As observed in Evan’s work, assumed delays in language are not necessarily a barrier that limits the growth of mathematical understanding. In fact, attending to and assessing this student’s thoughts, actions, and feelings, we can evaluate with confidence that Evan demonstrates spatial-geometric understandings that not only meet but, in certain instances, exceed prescribed expectations for the Kindergarten and Grade 2 provincial curricula.

This initial study and the insights from this research offer generative ideas for future work in deaf education, towards (re)conceptualising the live(d) ways young deaf children (re)(con)figure space aesthetically for STEAM education, especially in mathematics.