Enhancing mathematical noticing of graphs through movement,voice,and metaphor: An intervention with two students with visual impairment

Abstract

This qualitative study explores the potential for metaphor, movement, gesture, and vocalization in helping learners notice mathematically important features of graphs, and in making mathematics more accessible for learners with visual impairment. Two elementary school students with visual impairment were introduced to several multimodal activities related to the graphs of mathematical functions, using a pre-/post-assessment methodology. Video recordings of the session were coded for qualitative changes in engagement with graphs through multimodal representations. After the activity intervention, both students showed improvements in their ability to voice, gesture, and describe details of mathematical graphs with accuracy and understanding. The findings demonstrate the potential of multimodal methods for teaching mathematics and enhancing other skill areas through movement, metaphor, voice, and gesture. The findings suggest that full-bodied experience with graphs can provide foundational support for learners with visual impairment to work with print or tactile graphics. We propose that purposeful selection of materials and collaboration between teachers of students with visual impairment, mathematics educators, and teachers of dance and physical education can enhance the design and implementation of effective lessons using multimodal means.

Keywords

Accessible mathematics gesture graphs multimodal mathematics visual impairment

Introduction

For all students, mathematics learning involves being able to categorize, reason, and work on abstract mathematical objects (e.g., to treat complex conceptual entities like “functions” or “geometric transformations” as if they were concrete, manipulable objects). All of these actions depend fundamentally on noticing, attentiveness, focus, and awareness. Mason (2011) has researched and written extensively on the significance of noticing in mathematics education, and suggests that mathematical abstraction consists of “delicate shifts of attention” (Mason, 1989), and that attention can be guided and educated (Mason, 2002). Where novice learners may be focused on the particular properties of a mathematical example, teachers may be focused on the shared properties of whole classes of examples and abstract mathematical relationships. It is the teacher’s task to help direct learners’ attention to mathematically salient features, so that learners can begin to treat these as mathematical objects for reasoning (Sfard, 1994).

Metaphor, movement, gesture, and vocalization have potential for helping learners notice mathematically important features of graphs (Gerofsky, 2016; Lakoff & Núñez, 2000; Nathan et al., 2017). Different qualities of movement, gesture, and voice have been shown to have different effects for attending to mathematically salient features. Learners have demonstrated significant progress in mathematical understanding when large-scale gestures are integrated effectively into math lessons (Gerofsky, 2011). When learners have the experience of “being the graph,” rather than only “seeing the graph,” they have been shown to notice and reason with salient features with the greatest success (Gerofsky, 2010; Gerofsky & Ricketts, 2014).

In terms of metaphor in mathematics, Sfard (1994) writes, “Our conceptual system is a product of metaphors which transfer the bodily experience into the less concrete realm of ideas . . . Metaphor is what brings abstract concepts into being” (p. 46, emphasis in original). In a metaphor, the source (a concrete sensory experience) and the target (an abstract concept) are not absolutely identical, but the resonances or useful similarities between source and target help us think about abstract ideas in concrete terms (Fauconnier & Turner, 2008; Lakoff & Johnson, 2003). Previous research incorporating the use of metaphors to understand mathematical functions found that that metaphors are central to the expression and understanding of mathematical meaning (Pimm, 1981, p. 47), and vitally important in allowing learners to build new mathematical conceptions (Presmeg, 1998, p. 29). Conceptual metaphors and systems of metaphors form the basis for mathematical thought, including abstract reasoning and cognition (Núñez et al., 1999, pp. 51–52), and educators must engage with these metaphors to understand effective pedagogy and learners’ development of mathematical knowledge (Nuñez et al., 1999, p. 60).

Accessing and interpreting information from graphs can be challenging for students with visual impairments (Rosenblum & Herzberg, 2015; Smith & Smothers, 2012; Wall Emerson & Anderson, 2018; Zebehazy & Wilton, 2014a, 2014b). In particular, students with visual impairments who access graphs tactually may struggle to make sense of the whole or notice salient features since the tactile sensory system engages differently with information than the visual system. Students with visual impairments, in fact, may not recognize the benefit of graphics in helping their conceptual understanding (Zebehazy & Wilton, 2014c). While these studies did not specifically focus on graphs of mathematical functions, it can be hypothesized that the same challenges would exist as with other types of graphs. If learners with visual impairment are to succeed in secondary mathematics, it is important to find ways to enhance access to and understanding of the mathematical meanings of graphs and to engage in equitable pedagogy that fully engages their learning through sensory channels beyond vision (Abrahamson et al., 2019).

Success with older students with visual impairments engaging in embodied activities has shown promise for improving active engagement and student understanding of graphs. Figueras & Arcavi’s (2014, 2015) studies found that students were highly capable of noticing and integrating local and global spatial details of embossed graphs through touch and movement, and of combining kinesthetic experience and verbal metaphors to understand and communicate mathematical concepts (p. 185). Healy and Fernandes’ (2014) studies showed that embodied gestures are central to mathematical meaning-making for students with visual impairments, similar to the importance of sketching a diagram with a sighted learner, and multimodal embodied experiences support older students’ robust mathematical conceptualizations (Fernandes & Healy, 2013). Van Scoy et al. (2005) also found that a multimodal haptic/sonic computer interface has the potential to offer enhanced learning experiences of graphs for upper-level students with visual impairments.

Research questions

Our research extends previous research with students with visual impairments to explore whether younger students with visual impairment would benefit from a multimodal approach to understanding graphs they will later encounter formally in mathematics courses. Early and frequent engagement with graphs has been found to be a contributing factor to independence with graph reading (Zebehazy & Wilton, 2014a, in press). We hypothesize that a greater level of familiarity with these graphs and their significance may be helpful when such graphs are re-encountered later in school. Students with visual impairment have the challenge of trying to make sense of visual materials (graphs of functions) at the same time that they are learning the algebra related to mathematical functions in secondary school, and this often leaves them at a disadvantage. Our aim with this short intervention was to observe qualitatively any differences in how they approached unknown visuals/graphs, with the idea that such interventions at younger ages could support students being more readily able to access visual materials at the time when algebra is also presented.

To explore this approach, three collaborating researchers engaged two students in a half-day activity intervention as a theory-building, pedagogical design experiment (Cobb et al., 2003), using a pre- and post-assessment protocol, student interviews, video recordings of the intervention, and researcher reflective notes to observe changes and establish reliability of the observations. The specific research questions were as follows:

After a specific activity intervention, is there a change in the quality of movement and voice?

Are students able to create a variety of verbal, vocal, and movement metaphors to engage with the abstract forms of graphs?

Do learners’ accuracy and attention to the mathematically salient features of graphs change from pre- to post-assessment?

Do selected materials help or hinder engagement in the multimodal activities?

Method

Researchers

The researchers’ backgrounds were complementary and supported the study’s aims. The two authors are (1) a mathematics educator and (2) a special educator in the area of blindness and visual impairment. The third collaborator is an arts-based educator with expertise in dance/choreography. All three are research professors at Canadian universities. The study was approved by the authors’ university behavioral ethics research board.

Participants

Two students with visual impairment and no additional disabilities participated in the activities. The students’ parents provided informed consent and the students provided assent on the day of the activity. Kevin (pseudonym) was a Grade-4 male student with retinopathy of prematurity. He was reported by his teacher of students with visual impairment (TSVI) to have light perception in one eye and no vision in the other. In school, he had exposure to bar graphs. Carmen (pseudonym) was a Grade-5 female student with optic nerve hypoplasia (ONH), nystagmus, and exotropia. She has low vision and was reported by her TSVI to require 72-point font from a focal distance of one to two inches. She had exposure in school to bar graphs, pie charts, and had some experience plotting coordinates on tactile graph paper. Both students were braille users.

Setting

The intervention took place over the course of a half day in the large, open spaces of a drama and dance studio at a university in central Canada near the students’ elementary school. These were typical of other spaces used as part of the on-going work of the first author on embodied learning of graphs.

The students were accompanied by their TSVI and an educational assistant. Both worked with researchers to help students feel comfortable and to facilitate activities. Two undergraduate students acted as videographers to capture video data. A variety of materials were used to make the activities multi-sensory and tactile (see section “Intervention description”).

Session layout

The half-day session was conducted in the following manner:

Students were introduced to the researchers and project, and had time to explore the room.

Each student was interviewed individually about their feelings about mathematics.

In the pre-assessment, each student individually explored two tactile graphs (see Figure 1) and was asked to describe each graph, provide metaphors for the shape of the graph, gesture the graph, and voice the graph.

Students worked together with the researchers on intervention activities.

In the post-assessment, each student individually explored two different graphs with similar qualities to those in the pre-assessment (see Figure 1), using the same protocol.

Figure 1.

Pre- and post-assessment graphs.

Intervention description

Bodily warm-up: The warm-up was used to activate the body, build confidence and comfort with movement, and to establish movement metaphors that would later facilitate graph building. The researcher who was an arts-based educator led the warm-up. She verified knowledge from the two students of metaphors she chose, taking time to explain a metaphor if needed. Metaphors most accessible to the students were chosen to match with body movements. Symmetrical and asymmetrical movements, prompted through familiar gesture names (e.g., “catching a ball,” “smoothing a tablecloth”), were designed to ground students and work with both sides of the body. Students progressed to creating movement sequences, speeding up movement for fluidity, and to engaging the whole body in a playful, confident way. The warm-up was recorded to compare movements of the students during pre-test and the beginning of warm-up to the movements at the end of the warm-up, and during activities.

Establishing a mountain metaphor for the curve of a graph: After warm-up, students focused on a mountain metaphor, gesturing “going up and down the mountain” from the floor on one side of the body to the floor on the other side of the body, using both arms and engaging the spine. Rising and falling vocal pitch were connected with “going up and down the mountain.” The activity focused on the relationship of movement and voice, connecting the metaphor to the curves they would later explore in polynomial graphs.

Calibrating the height of a curve using the mountain metaphor: Students learned to use numbers from 1 to 10 to calibrate their height on the “mountain” or graph. They connected this numerical calibration with vocal pitch and gesture so that they could name a particular height as a seven, a four, and so on. The activity drew further attention to the connection between quality of movement and voice, features of a curve, and resonance between the “mountain” metaphor and the graph.

Preparing to build a graph: Since mathematical graphs include more than one “mountain,” the metaphor was expanded to “traveling a mountain range.” Students began by working with one of the researchers and then with one another, facing their partner with palms touching, moving and voicing in tandem to represent several graphs. An elastic cord was introduced to represent the x-axis, and the sound “pah” was used to indicate crossing the x-axis. Crossing the x-axis is mathematically significant to draw attention to the roots of the function. The activity built teamwork, helped students translate from voice to movement, and drew attention to features of the graphs.

Building a graph at different scales and with different materials:

(a) Activity 1: One student–researcher pair moved and voiced a graph, while the other student followed the vocalized graph shape with their finger along a length of a table, with plastic tubing representing the x-axis. When the student–researcher pair vocalized a root of the function with the sound “pah,” the other student checked that they were crossing the plastic tubing at that same moment, or adjusted accordingly. Roles were then switched.

(b) Activity 2: One student–researcher pair moved and voiced a graph as above, while the other student drew what was being voiced using plastic film on a rubber mat with a stylus. The voicing student then checked the drawn graph. They discussed and then switched roles.

(c) Activity 3: A “monster polynomial” was created using thick yarn placed on the carpeted floor in the shape of the polynomial graph, using a heavier cord as the x-axis. Each student crawled along the graph, voicing as they went along. They voiced “pah” as they crossed the x-axis.

Data analysis

The co-authors first watched video of the whole session individually, recording reflection notes of their observations of pre-assessment, intervention, and post-assessment. The researchers then engaged in an in-depth conversation through the process of repeated viewing and coding of video data and field notes together, using an emergent grounded theory paradigm (Charmaz, 2008) to discover observable changes between pre- and post-assessment engagement of the students in relation to the research questions. The researchers found the following coding scheme to be a good fit to the data, with a high degree of inter-rater agreement established through this process:

Quality and scale of movement (engagement and accuracy in using small and large body movements to represent graphs);

Quality of voice (engagement and accuracy in vocal representations);

Metaphor (selection of metaphor related to the graph and level of attention to the resonance between the metaphor and the graph);

Noticing (paying attention to mathematically salient features);

Learning (development of more robust skills using multimodal means);

Interaction of material choices.

Results

Research question 1: quality of voice and movement

Kevin

In the pre-assessment, Kevin displayed no hesitation in using his whole body in a very kinetic way. He started by working in the horizontal plane of the table top or floor, gesturing with his arm and whole body (locomoting the path represented by the graph), remaining in the same plane when translating from feeling the tactile graph to gesturing it. This was in contrast to the more conventional way of gesturing graphs in the vertical plane. He also lay down on the floor to move like a snake, one of his metaphors for the curve of Graph 1 (see Figure 1 for all graphs). For Graph 2, Kevin walked a path on the floor, putting himself into the graph, and turned right and left alternately to create a representation of the curves of the graph, but he extended the graph to more curves than were in the actual representation. This approach of “being” or “riding” the graph is congruent with observations from earlier phases of the project where learners who were considered top mathematics students used similar gestures and metaphors (Gerofsky, 2010). When Kevin first voiced Graph 1, he clearly marked the turning points with a pause and change in pitch direction. However, in this initial phase, the choice of rising or falling pitch was unrelated to the rise or fall of the graph. The same pattern was observed with Graph 2.

In the post-assessment, Kevin, as in the pre-assessment, did not always link voice pitch to the direction of the curves; however, there was a subtle change from the pre-assessment in that he self-corrected or restarted to better match his voice to the graph. On Graphs 3 and 4, Kevin initiated voicing without researchers prompting him to do so. He would also pause and self-correct and added “pah” at the correct locations. We observed a clear intention to match his voice to the curve and an increasing ability to control the voicing. In addition, during the activities, he was observed to be quite accurate in following other people’s voices when doing small-scale gestural drawings of what he was hearing. Unfortunately, due to constraints of time and attention, Kevin did not gesture the post-assessment graphs.

Carmen

In the pre-assessment, Carmen initially had a more difficult time representing the tactile graph accurately through gesture. Rather than gesturing the curves, her arm movements (in vertical plane) included more turns than were in the original graph, and the gestures had a circular, rather than a linear quality. She used whole arm movements and locomotion, but at the start, the representations were only loosely related to the tactile graphs. Toward the end of the pre-assessment, Carmen began to give more attention to the form of the graph and to gesture more accurately. Like Kevin, Carmen used whole body movements from the start. For example, when gesturing Graph 2, even when seated, she reached high above her head and also engaged her head and neck in the movement. When asked to voice the graphs, Carmen was reluctant to do so in the pre-assessment. When asked to describe what the sound would be for the researcher to voice for her, she focused on increasing speed of voice rather than pitch related to the curves of the graph.

In the post-assessment, Carmen readily voiced the graphs when requested. When voicing, she matched her tone and pitch to the graph and indicated “pah” when crossing the x-axis in both directions. She showed no hesitation to voice, in contrast to the pre-assessment. When voicing the second graph in both directions, she matched her pitch to location of her fingers on the graph and made self-corrections. When asked to gesture the post-assessment graphs, Carmen showed a distinct change from the pre-assessment. When asked to gesture Graphs 3 and 4, she checked the shape of the graph carefully and gestured using full arms with both hands together. Her gesture of the graph in both directions was controlled and accurate. On the first pass she showed confidence in putting herself a bit off balance, and on the second pass she improved the accuracy and engagement and engaged her spine in the movement as well. When asked to gesture the second graph, Carmen confidently responded, “I think I can do that” and used full arm motions and movements, engaged her spine, and went up on her toes to gesture the graph in both directions. This showed a marked change from the pre-assessment where her movements were only loosely related to the graphs.

Research question 2: use of metaphor

Kevin

In the pre-assessment, Kevin chose and persisted with the metaphor of a snake or worm for Graph 1. His movements and use of a hissing sound were connected with his strong attachment to the source domain for this metaphor. In Graph 2, he related to “holes” in the graph, which were the minimum points in the curves.

During the post-assessment, Kevin identified Graph 3 as a path with one hole and two curves. When using a toboggan metaphor to follow the path, he stated that the toboggan would fall into the hole. In Graph 4, without prompting for a metaphor, he related the shape to a snake, similar to his earlier metaphor in the pre-assessment with a different graph. When asked if the snake looked the same or different from the other snake from the pre-assessment, he accurately indicated that it was different. When asked to consider the same graph as if it were a path, Kevin was able to indicate two “hills” and point them out.

Carmen

In the pre-test, Carmen produced several metaphors, with a focus on the source of the metaphor (from experience or stories) rather than the resonance between the story and the graph. As indicated in quality of movement, her movements acted out the story or metaphor rather than gesturing the actual shape of the graph. In Graph 1, she referred to two stories about pathways: sledding down a hill on the school playground and her cousin running through the house. In Graph 2, she referred to a toddler scribble, and her gestures indicated many more curves and bumps than were present in the graph.

In the post-assessment, Carmen also selected metaphors related to personal experience. For Graph 3, she related the shape to a hallway – specifically the straight hallway she just walked at the university, even though this was not a good fit with the graph. Whatever experience came to mind, Carmen seemed to be “retrofitting” it to the graph she was exploring. On Graph 4, she related the shape of the graph to the way she used to make her print “m’s” shorter on one side than the other. This showed improved accuracy and attendance to the actual shape of the graph as she noticed the asymmetry of the graph. The second metaphor she provided related to a story about her cane making a zigzag in the snow as she walked. As in the pre-assessment, Carmen related the graph shapes to stories, but with greater control and greater attention to the features of the graph. She was more willing to stay with the resonance of the metaphor rather than abandoning the graph and engaging purely in the story.

Research question 3: noticing and learning

Kevin

In the pre-assessment, Kevin readily explored the graphs. Using two hands, he went back and forth over Graph 1 several times, left to right and right to left. He was also able to notice the rotational symmetry by turning the graph 180 degrees and seeing that it was the same shape. With Graph 2, he noticed that it was different from Graph 1 and explored it with both hands in both directions several times. He was able to notice symmetry and asymmetry in the graph. As noted earlier, in walking the graph, Kevin paused at maximum and minimum points and gave attention to them. In making sharp turns, he spiraled the core of his body to feel/express how extreme the curve was. He also noticed that the graph had curves and “holes” and that the curves were of different heights. He could count the number of curves and “holes.”

In the post-assessment, Kevin showed interest in the components of the graphs and what materials they were made of. On Graph 3, Kevin gave a more accurate verbal description of the shape of the graph, compared to the pre-assessment, where he was more interested in the image of the snake (e.g., hissing and slithering on the floor) rather than the resonance between the snake metaphor and the shape of the graph. As in the pre-assessment, he identified the curves and “holes” of Graph 4 and noticed that the hole was “deep.” Using two hands (one hand on each curve), he was able to identify that the curves were of different heights.

Carmen

In the pre-assessment, Carmen could describe the shape of Graph 1, but as soon as metaphors were introduced, she moved away from that description into the story. For example, when asked where the sled in her metaphor would end up on the “hill,” she indicated a location off the graph which represented an imagined hillside from a three-dimensional (3D) perspective. Her attention was focused on other features of her story that extended beyond the edge of the paper. Carmen also reoriented the graph to facilitate her storytelling. When indicating how the researcher should voice the graph, Carmen pointed with her hand and used deictic language (e.g., “here”) to indicate the maximum and minimum points of the graph. On Graph 2, she noticed the addition of the x-axis and asked what it was. Carmen gave an accurate description of the graph using words like “straight part here,” “then it goes curvy,” and “stops.” She also indicated the relative sizes of the curves. After multiple attempts and prompts by the researchers to focus Carmen’s attention to the shape of the graph, Carmen gestured an accurate depiction of Graph 2 moving in both directions.

In the post-assessment, Carmen’s exploration with her hands and verbal descriptions were much more attentive to mathematically salient features of the graphs. With both Graphs 3 and 4, Carmen’s first move was to notice the relationship of the graph to the x-axis and comment on it. In terms of mathematical noticing and learning, this was quite significant as her attention was not only to the graph but also to its context and positioning in the plane. On Graph 3, she noticed that the graph went above and below the “pah” line (x-axis) and that the line was diagonal with a slight curve. She could also tell the difference between this graph and the graphs created during the intervention activities and could describe why they were different. Before gesturing, she took time to double check the details of the graph, and rechecked the graph before gesturing in the other direction, showing attention to detail and pattern. On Graph 4, she showed an integration of the different modalities, noticing that the graph did not cross the tangible x-axis and that therefore her voice should remain at a high pitch throughout. She now had several different ways of representing that graph through tangible media, voice, movement, and metaphor. Carmen demonstrated an ability to make a seamless transition from one equivalent representation to another, an ability that could be equated with depth of mathematical understanding (Pape & Tchoshanov, 2001). As with Kevin, she noticed that the curves of the graph were not symmetrical and incorporated that into her metaphor (drawing an asymmetrical “m”). When preparing to voice, Carmen checked the x-axis and self-corrected her pitch. She showed self-awareness and attention to the graph in a way quite distinct from the pre-assessment. As with Graph 3, she checked the details before gesturing and used sub-vocalizations to help her keep track of the shape of the graph. Again, we saw a striking change and increased ability to integrate different multimodal representations of the graph.

Research question 4: materials

Kevin

Kevin was keenly interested throughout the process in the materials used to create the pre- and post-assessments and activities. At times, the materials distracted Kevin from focusing on the larger intent of the activity, that is, understanding graphs. Moments when we allowed Kevin to explore the materials and explained how they worked were successful in then refocusing him on the activity.

Carmen

Carmen showed less specific interest in the materials used for the activity, and engaged with them for the intent of the activity without much comment.

Both

It was observed that some of the selected materials and activities worked better than others. In Activity 1, the thickness of the plastic tubing posed some challenges for Kevin and Carmen to cross the tubing and keep up with the rate of vocalization. In Activity 2, the use of the plastic film was successful in having something tangible for the student voicing to explore and reflect on, but both students needed some support to use the stylus and keep up with the rate of vocalizations. In Activity 3, the use of the wide yarn in a very large-scale format was observed to be the most successful in terms of student independence and facility with the materials.

Discussion

Summary of results

In revisiting our four research questions, we found that,

The quality of movement and voice changed observably for both students from pre- to post-assessment. The change was most dramatic for Carmen. In the pre-assessment, her movements were not always closely linked to the actual shapes of the graphs and she refused to voice. By post-assessment, she was voluntarily voicing and gesturing in an integrated way, with sustained attention to the shape of the graphs. With Kevin, we noticed self-correction as part of a continued effort to match the pitch of his voice to the shape of the graph. While in previous studies (Gerofsky, 2010) some students tended to use smaller, more distal gestures, both Kevin and Carmen were willing to use whole body movements and large gestures from the start. Carmen showed a change after the intervention and began using movement engaging her whole spine.

In terms of metaphor, both students were willing to produce some metaphors in the pre-assessment but were uncertain about the relationship of these metaphors to the detailed shapes of the graphs. Both began by focusing more on the source domain of the metaphor to the exclusion of the resonance between the metaphor and the target domain (shape of the graph). Carmen in particular took the discussion to the realm of story, and in the pre-assessment, got lost in the story. She showed the most difference in making a switch to attending to the resonance between source and target domains by post-assessment. Kevin applied similar metaphors to the new graph shapes in the post-assessment, but was more attentive to the ways the metaphor would play out with the precise shape of the graph (e.g., toboggan falling into the “hole”). He used gestures along the graph, synchronized with his words, as an aid to storytelling. Post-intervention, both students could be observed using metaphors as ways of pointing to particular abstract features of the graph.

In terms of attention to mathematically salient features of the graphs, we saw marked improvements in both students. Carmen in particular showed an ability in post-assessment to integrate various multimodal representations as part of an imaging of the graph through sound, gesture, movement, metaphor, and tactile graphics. All these elements came into play along with a much greater attention to the mathematically important features of the graph itself (e.g., roots, slope, curvature, relative height and number of maximum and minimum points). She seemed better able to describe and compare the graphs of polynomial functions with accuracy and discernment. Kevin was more accurate in the post-assessment in describing the detailed features of the graphs and was able to integrate voice and metaphor somewhat better after the intervention. Although his progress was not as dramatic after one intervention session, we did see his attentiveness to mathematical features of graphs was enhanced to some degree. For example, like Carmen, he integrated voicing the locations where the graphs crossed the x-axis (“pah”) in a way that seemed effortless and natural.

For students who are visually impaired, it is important to consider materials that allow for independent manipulation and discrimination of tangible features. For example, it was important for us to use a different material for the x-axis (e.g., plastic tubing, elastic) from the graph itself. This highlighted mathematical noticing and allowed for the integration of other multimodal representations (e.g., saying “pah” as the graph crossed the x-axis). Some experimentation with a variety of materials may be needed. For example, the use of the plastic film and stylus for drawing the graph being voiced (see item 5(b) in section “Intervention description”) turned out to be challenging for the students to execute independently because the stylus would get caught in the film, making it hard to keep up with the voicing. While it is beneficial to be able to go back and review a drawn graph, a better medium is needed. The plastic tubing, while quite noticeable, may have been too high off the table surface, causing some difficulties in crossing it as an x-axis.

Benefits and potential application to instruction

Students who are blind or visually impaired may struggle with understanding graphs and diagrams (Rosenblum & Herzberg, 2015; Smith & Smothers, 2012; Zebehazy & Wilton, 2014a, 2014b, 2014c). In addition to the provision of descriptions to support understanding of graphs, the results of this study indicate there may be a positive effect of engaging younger students in multimodal activities to facilitate understanding of mathematical representations through graphs they will later encounter in future math courses. The integration of large- and small-scale movements with voice and metaphor seems to offer a clear benefit for developing a deeper understanding of concepts related to mathematical graphing. This may translate to other abstract conceptualizations within in mathematics or other areas of study. For example, geometric concepts, ideas about mapping, numerical relations, and timelines could be represented through the use of movement, voice, gesture, and metaphors. This may help students grasp the concepts at a deeper level.

Attention to the instructional design around these representations is essential to their success for student learning. When designing such an intervention, it is important to make sure that the movement, voice, and metaphors introduced by teachers are truly analogous to the concepts being targeted, so that they focus attention on the learning goal. Our collaboration across different disciplines and research areas served as a way to check that we were designing both an accessible and focused intervention. We recommend similar collaborations among teachers of students with visual impairment, subject area teachers, and drama, dance, or physical education teachers.

The use of gesture, movement, and voice had another beneficial effect. During the warm-up, the incorporation of large body movement and metaphor practice supported the students’ ability to engage in these movements with ease and minimal physical guidance, which was an observed change in the videos from the movements made by the two students during the pre-test and at the very beginning of the warm-up. The development of a movement repertoire in this way might support the instruction that teachers of students with visual impairment undertake in areas of the expanded core curriculum such as recreation and leisure. A general confidence in using movement and voice may potentially bring additional long-term benefits.

Limitations and future research

This study only involved two participants and, thus, cannot necessarily be generalized. However, the results of these young students with visual impairment coincide with findings from the first researcher’s graphs and gestures research project over a 10-year span, working with approximately 150 students, including 4 older students with visual impairment. Carmen and Kevin were the youngest students with whom these methods have been tried. This suggests that the multimodal approach shows promise of having wider benefits for students with visual impairment and with learners generally. Future research should include more students with visual impairment at different grade levels as well as continued attention to identifying multimodal activities and accompanying materials that provide optimal accessibility and the possibility of greater agency for students with visual impairment to be actively engaged in mathematical learning. This study did not return to the students to assess lasting learning, which could be a component of a future study. The idea of extending beyond the graphs of mathematical functions to other graphical representations of abstract ideas may also be worth pursuing.

Footnotes

Acknowledgements

The authors thank their collaborator, Kathryn Ricketts, PhD, University of Regina, for her contributions to this study.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by a grant from the Canadian Social Sciences and Humanities Research Council (SSHRC).

ORCID iD

Kim T Zebehazy

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