Exploratory Study on Geometric Learning of Students with Blindness in Mainstream Classrooms: Teachers’ Perspectives Using the Van Hiele Theory

Abstract

Ensuring mathematics education for all learners, including students with blindness learning in mainstream classrooms, is crucial. This exploratory research aims to clarify the characteristics of geometric learning among students with blindness and to identify the factors contributing to the challenges faced by this population. The Van Hiele theory of geometric thought served as a reference framework. Qualitative data were gathered through group interviews with specialists in the field of education for students with blindness and analyzed using inductive analysis. Participants affirmed that students with blindness progress through Van Hiele levels of geometric thought in a manner similar to sighted students, suggesting that much of the learning can take place alongside sighted peers in mainstream classrooms. However, they also highlighted the unique challenges these students face in reaching level 0, a level where students recognize shapes without a formal understanding of their properties or attributes. Among the reasons for these challenges were that for these particular students, subskills, such as bimanual exploration, hand coordination, and cognitive integration, are required to reach level 0. The study also identified the necessity for specialists in visual impairment education to guide students using appropriate tasks and learning materials that reflect the characteristics of haptic perception. Since level 0 serves as a gateway to both basic and advanced geometry, the findings underscore the importance of providing differentiated support that targets these subskills early in students’ schooling. To ensure meaningful geometry instruction, mainstream teachers are encouraged to collaborate with specialists in visual impairment education, who can guide the selection of appropriate learning tools and support the development of the subskills.

1. Introduction

Geometry, a branch of mathematics addressing spatial sense and geometric reasoning (Howse & Howse, 2014), is a fundamental component of education. Compared to other mathematical topics like numbers, algebra, measurement, and data analysis, geometry occupies a considerable portion of the curriculum at every educational level (Trimurtini et al., 2022).

Blindness is a condition within the spectrum of visual impairment, characterized by a decrease in the ability to see to a degree that challenges daily living or access to learning and cannot be corrected with glasses or contact lenses (World Health Organization, 2019). Students with blindness predominantly use braille and tactile materials for learning. Their needs are distinct from those who are sighted or those who have lost their vision later in life, as these groups can rely on visual imagery, whereas congenitally blind individuals depend on haptic imagery—imagery perceived through touch. Students with blindness, whose impairment is solely in vision, are expected to learn the same mathematics curriculum as their sighted peers and achieve equivalent outcomes. Therefore, providing adequate geometry education to students with blindness is equally important as it is for students without blindness.

In recent years, there has been a growing trend to include students with disabilities, including blindness and other diverse needs, in mainstream rather than specialized schools (e.g., schools for the blind). This shift toward inclusive education is influenced by international and national frameworks, such as the Convention on the Rights of Persons with Disabilities (United Nations, 2006) as a former example. These frameworks assert that all students, including those with blindness, have the right to be included and receive adequate education alongside their sighted peers in their local communities or mainstream schools. This right to inclusive education has opened new avenues for individuals with disabilities and prompted mainstream schools to become more flexible and innovative in addressing the needs of diverse learners, many of whom had previously been overlooked (Dalgaard et al., 2022; Szumski et al., 2017). However, while inclusive education is promising in theory, in reality, students with specific disabilities, such as blindness, face many barriers in mainstream schools. A key example of this is their exclusion from higher-level “visual subjects”—subjects that rely heavily on visual input or practical applications, such as mathematics, science, and physical education (Miyauchi, 2020). Despite their abilities, students with blindness are often denied access to these subjects, highlighting a significant shortfall in inclusive education.

The Van Hiele theory of geometric thought, which this research used as a reference, is a well-known framework explaining how students learn geometry (Crowley, 1987). It consists of five levels of understanding, beginning with level 0 (visualization), where students recognize shapes by their visual appearance without a formal understanding of their properties or attributes. At level 1 (analysis), students notice different shapes by recognizing attributes but do not yet understand their relevance. This is followed by level 2 (abstraction), level 3 (deduction), and level 4 (rigor), where students progressively identify and analyze geometric forms, moving toward higher levels of abstraction (Crowley, 1987; Naufal et al., 2021). Past studies confirmed that students transition gradually from one level to the next, often moving back and forth between stages without sudden jumps or skipping levels (Duroisin & Demeuse, 2015).

To the authors’ knowledge, Argyropoulos (2002) is the only study that explores Van Hiele’s theory in relation to students with blindness. This study suggests that Van Hiele’s theory is a suitable framework for examining the geometrical thinking process of these students while also highlighting the unique challenges they face, particularly at the earlier levels. However, given the scarcity of research on this topic, further investigation is needed.

Exploring how students with blindness learn geometry and the challenges they face using a framework designed for sighted learners is crucial for advancing inclusive education in the following ways. First, this process, we believe, will help reveal both the similarities and differences in how children with and without visual impairments learn. Recognizing these similarities and differences is essential, as societal perceptions of human differences can be both positive and negative—what some researchers term the dilemmas of difference (Norwich, 2008; Paulsrud, 2024). Acknowledging and addressing a child’s unique learning needs may lead to physical separation (or segregation), potentially resulting in feelings of exclusion and a lack of acceptance among peers. However, failing to recognize and respond to these differences can limit students’ access to resources and specialist services, hindering their equitable participation in education and society. Hence, finding the right balance between differences and similarities in how children learn is crucial.

Second, this process helps us understand what effective differentiated teaching within a mainstream classroom may look like—one that strikes the balance between differences and similarities in how children learn. Differentiated teaching is a method employed by teachers that involves assessing and monitoring students’ learning readiness and processes (Vaughn et al., 2022/2023). Its purpose is to extend the knowledge and skills of each individual child; hence, this approach includes small-group or individualized instruction and the use of tailored instructional tools, rather than rigid whole-class instruction with the same tools for all students. Incorporating differentiated teaching into mainstream classroom instruction has been identified as a key strategy for enhancing inclusion (Soan & Monsen, 2023).

This research uses the Van Hiele theory of geometric thought to explore how students with blindness learn geometry. This approach provides valuable insights into achieving a balance between recognizing commonalities in the learning processes of students with blindness and sighted students and identifying the specific content that should be reflected in differentiated teaching. Such an understanding can guide mainstream teachers in effectively collaborating with specialists, supporting and enhancing the learning experiences of students with blindness in classrooms.

This study aims to clarify the characteristics of geometry learning among students with blindness in comparison to their sighted peers, by interviewing specialists in the education of students with blindness using the Van Hiele theory of geometric thought as a reference. By highlighting both similarities and differences, this research seeks to identify areas where differentiated teaching is needed to foster effective inclusive education. Furthermore, it aims to provide practical insights for mainstream teachers to enhance the learning experiences of students with blindness.

The specific research questions addressed are as follows:

• Does the Van Hiele theory align with how teachers and specialists observe their students with blindness learn geometry?
• What specific challenges do students with blindness face in learning geometry?
• What are the origins of these unique challenges?

2. Materials and Methods

2.1. Study Design

Given the limited research on geometry education for students with blindness, this study adopted an exploratory design. The research focused on qualitative group interviews with a purposive sample of Japanese individuals who possessed extensive knowledge of teaching both students with and without blindness. Qualitative research methods were chosen as they are appropriate to expand knowledge on unexplored areas. Group interviews were chosen owing to their ability to foster dynamic discussions, allowing participants to build on each other’s responses and generate rich data through group interaction (Krueger & Casey, 2014).

2.2. Study Setting

2.2.1. Study Context

With the Japanese government promoting an “inclusive education system”, whereby students with disabilities are enrolled in mainstream classrooms to the extent possible (Ministry of Education, Culture, Sports, Science and Technology, 2013), more children with visual impairment are learning in mainstream settings. However, within visual impairment, students who uses braille (hence, students with blindness) tend to continue to enroll in schools for the blind, as these schools offer more specialized support. Japan is a unitary state where the central government, specifically the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), governs the national curriculum known as the Course of Study. All textbooks used in schools are authorized by MEXT based on the Course of Study. Students with blindness are required to follow the same Course of Study as their sighted peers. Hence, braille textbooks for these students are created by modifying textbooks used in mainstream schools.

2.2.2. Recruitment and Sampling

Purposive sampling was employed to recruit participants with extensive teaching knowledge who could provide in-depth insights into the similarities and differences in geometry learning between sighted students and students with blindness. The selection was based on two main criteria. First, participants had to be either teachers or lecturers. Teachers were required to hold teaching certificates in mathematics or other relevant subjects (e.g., geography) for both mainstream schools and schools for the blind, with at least 20 years of teaching experience. Lecturers, on the other hand, needed to have a PhD in the field and demonstrate expertise through teaching and research on the education of students with visual impairments. Second, to ensure participants had extensive knowledge of the similarities and differences in how students with blindness learn compared to sighted students, all participants—whether teachers or lecturers—had to have served as MEXT-appointed editorial board members in adapting school textbooks into Braille versions for students with blindness. In Japan, braille modification is conducted strictly to preserve the content of mainstream school textbooks, altering only the aspects that must be adapted due to vision impairment. As a result, editorial board members are required to have expertise, not only in how blind students learn, but also in how sighted students learn, ensuring that the adapted materials remain as equivalent as possible. This selection process ensured that participants aligned with the purpose of this research. Lastly, participants who were blind themselves were intentionally included if they met the above criteria. This approach ensured that the study incorporated the lived experiences of blind individuals, who learned through tactile perception.

A total of five teachers and experts in the field participated in this study. Of these, four were teachers (three math teachers and one geography teacher) and had an average of 27.75 (SD: 6.0) years of experience teaching students with blindness. Among the four teachers, one was female and three were male. The three male teachers were congenitally blind. All four teachers held dual teaching certificates: one for lower and upper secondary school (subject areas of mathematics or social studies, geography, and history) and another teaching certificate for special needs education (with a focus on visual impairment). All were actively involved in editing braille textbooks authorized by the government in addition to training other educators and authoring books on the education of students with visual impairment. One was a part-time university lecturer who was congenitally blind and possessed expertise in tactile arts and graphics for students with blindness. Participant details are described in

Table 1. Demographics of the individuals interviewed.

Participant	Sex	Age	Occupation	Obtained Degree/Licensure	Number of Years Teaching	Visual Impairment Status
A	Male	60s	Lower and upper secondary math teacher at school for the blind, retired	Teaching certificate for lower and upper secondary school mathematics/teaching certificate for special needs education (visual impairment)	36	Blind
B	Male	50s	Lower and upper secondary math teacher at school for the blind	Teaching certificate for lower and upper secondary school mathematics/teaching certificate for special needs education (visual impairment)	28	Blind
C	Female	50s	Lower and upper secondary math teacher at school for the blind	Teaching certificate for lower and upper secondary school mathematics/teaching certificate for special needs education (visual impairment)	25	Sighted
D	Male	50s	Lower and upper secondary social studies, geography, and history teacher at school for the blind	Teaching certificate for lower and upper secondary school social studies, geography, and history/teaching certificate for special needs education (visual impairment)	22	Blind
E	Female	60s	Parttime university lecturer	Doctor of Philosophy in art	NA	Blind

.

2.2.3. Data Collection and Ethical Considerations

A total of four in-person group semi-structured group interviews, with all participants present were conducted. The number of group interviews was determined based on data saturation. Each interview lasted approximately two hours.

A semi-structured interview protocol was developed specifically for this study based on the research question. The protocol consisted of five open ended questions: “What are your thoughts on the Van Hiele theory? In what ways does Van Hiele theory align with or differ from students with blindness’ geometric learning?”; “What are your thoughts on the levels? Does learning typically proceed through these levels?”; “What challenges do students with blindness face in learning geometry?”; “How do you address these challenges as a teacher?”; and “What factors do you think contribute to these challenges?” During the initial meeting, the Van Hiele theory was introduced and explained to participants, along with definitions of each level. Participants were encouraged to express their thoughts freely so that the conversation flowed naturally. The lead author facilitated the interviews and used the open-ended questions as a guide to ensure discussions remained relevant and did not stray from the research focus.

Prior to the interviews, participants were provided with a consent form, which the first author read aloud, and written consent was obtained. All interviews were audio-recorded, transcribed verbatim, and anonymized during transcription. This research was approved by the Research Ethics Committee of the first author’s affiliated university (No. Tsuku2023-232A).

2.2.4. Analysis

The data were analyzed inductively with the help of NVivo software, following the guidance of Braun and Clarke (2006). This approach involved reading and re-reading the data to identify initial ideas, systematically coding the entire dataset, and collating these initial codes into potential themes. The first author, who has PhD training in qualitative methods, performed this process. The themes were then reviewed with the co-author to ensure their applicability across the entire dataset. Subsequently, the themes were refined, named, and representative extracts were identified as exemplars for each theme. This iterative process continued until thematic saturation was achieved.

3. Results

3.1. Results from the Analysis

The following four themes emerged from the analysis: “similar geometric thought processes as sighted individuals”; “challenges exclusive to students with blindness at the visualization level”; “visualization level requiring multiple tactics”; and “the need for specialists to guide students with blindness with appropriate tasks and learning materials”. Each theme is summarized in

Table 2. Themes and illustrative data.

Theme	Illustrative Data
Similar geometric thoughts processes as sighted individuals	“Yes, this (levels described by Van Hiele) makes sense. I agree with the levels described by Van Hiele” (Participant B). “The process (described by Van Hiele makes sense” (Participants A, C, D, E).
Challenges exclusive to students with blindness at the visualization level	“Once the blind students can pass this level, the other levels will be just like for the sighted” (Participant B). “Level 0 is where you will see the challenges among students because they have no vision to rely on” (Participants A, C).
Visualization level requiring multiple tactics	“I have seen many children who can ‘touch’ but cannot understand what they are touching” (Participant C). “Good hand movement, along with the skills to obtain and integrate tactile details in their head to understand the overall shape or form of an object, is necessary… because you have no vision to rely on” (Participants A, B). “Whether or not the blind child is interested in touching matters a lot” (Participants B, D, E).
The need for specialists to guide students with blindness with appropriate tasks and learning materials	“Our hands naturally curve, making it easier to perceive 3D shapes. However, with 2D shapes, one must intentionally control the fingers, adjusting both pressure and direction to accurately perceive the shape” (Participant A). “When touching a contour line, the child needs to be able to distinguish curved line from a straight line, which is hard for blind children who cannot rely on vision…we usually have blind students practice using shapes with curbed line, like a circle… and straight line, using like a polygon, first so they understand the difference” (Participants A, B, C).

, along with illustrative data. The following sections provide a detailed explanation of each theme.

3.1.1. Similar Geometric Thought Process as Sighted Individuals

All participants agreed that students with blindness follow the geometric thought process outlined by Van Hiele. Furthermore, they confirmed that although students may go back and forth between levels, there is a distinct thinking/learning sequence. Therefore, level 0 serves as the foundational level necessary for students to progress to levels 1, 2, and beyond, echoing past studies focused on sighted students.

3.1.2. Challenges Exclusive to Students with Blindness at the Visualization Level

While all participants agreed that students with blindness follow the thinking process suggested by Van Hiele, they particularly emphasized the importance of the level 0, the visualization level. Unlike sighted students, who typically progress through this level with ease, some students with blindness encounter significant obstacles, preventing them from advancing beyond visualization and causing delays in understanding more advanced geometry. Participants remarked, “Once the blind students can pass this (the visualization) level, the other levels will be just like for the sighted. Of course, some students will have difficulty transitioning from the analysis level to the abstraction level, but that is similar to sighted students, and the reason would not solely be due to vision” (Participant B).

3.1.3. Visualization Level Requiring Multiple Tactics

All participants highlighted the significant differences and complexities that students with blindness face at level 0, particularly concerning visualizing shapes through touch. One teacher emphasized, “I have seen students who can ‘touch’ but cannot understand what they are touching. If you are sighted, you see and you understand what you are seeing” (Participant C). Another teacher, who is blind, added, “The act of touching does not equate to the act of seeing” (Participant A).

According to participants, visualizing shapes through touch requires a range of complex skills and strategies, even if the shape is a simple, basic geometric form. These challenges highlight the intricate haptic system that students with blindness must navigate because of their impairment. For instance, participants noted that for children with blindness to visualize geometric forms, the ability to effectively use both hands to capture the object as a whole—and not just a portion—is foundational. While sighted students typically perceive an entire object at once unless parts are covered, students with blindness may touch only a portion of an object and may misidentify it. One teacher shared, “A blind child who lacks basic hand exploration skills may touch only a portion of a rectangle, like the upper right corner, and mistakenly identify it as a triangle” (Participant C).

Effective hand strategies are also crucial. Participants emphasized the need for techniques where one finger remains on a fixed point while the other hand explores the overall shape, aiding the child in building a complete mental image. Equally important is the ability for students to distinguish fundamental features such as straight lines, curved lines, and angles after mentally assembling the image. Two participants explained: “We first practice distinguishing between curved and straight lines before introducing shapes like circles and polygons. This helps students understand the differences within these shapes when touching” (Participants B and C).

Furthermore, strategies for retaining and integrating the images obtained through touch were highlighted. Participants noted, “Verbalizing is essential for categorizing the information touched, allowing students to piece together the complex image in their minds” (Participants A, B, and C).

Lastly, intrinsic motivation also played a key role, highlighting the active nature of tactile exploration compared to the passive nature of visual perception. Participants expressed, “Whether or not the blind child is interested in touching matters a lot” (Participants B, D, and E). Another participant elaborated, “When using vision, external information comes automatically and unintentionally. When you are blind, information only comes in when you touch actively and intentionally” (Participant A).

3.1.4. The Need for Specialists to Guide Students with Blindness with Appropriate Tasks and Learning Materials

The size of the object and whether it is two-dimensional or three-dimensional also appeared to significantly affect the ability of children with blindness to reach level 0. Larger objects that exceed the size of a child’s palm presented additional challenges, requiring more advanced dual-hand manipulation skills.

All participants agreed that touching two-dimensional shapes requires more technique than touching three-dimensional shapes. One teacher explained: “Our hands naturally curve, making it easier to perceive 3D shapes. However, with 2D shapes, one must intentionally control the fingers, adjusting both pressure and direction to accurately perceive the shape” (Participant A).

All participants, who also taught students with blindness daily, mentioned how they supported them in developing these skills and tactics. Specifically, they provided adequate two-dimensional and three-dimensional models and offered verbal cues on how to tactically touch and retain tactilely perceived information in their minds.

4. Discussion

This exploratory research aimed to clarify the characteristics of geometric learning among students with blindness and to identify the factors contributing to the challenges faced by this population. Through interviews with specialists in the education of students with blindness and using the Van Hiele theory of geometric thought as a framework, it sought to provide practical insights for mainstream schoolteachers on how they could effectively support and enhance the learning experiences of students with blindness.

Participants confirmed their agreement with the structure and processes outlined in the Van Hiele theory, aligning with previous research by Argyropoulos (2002). The findings suggest that the geometry learning process for students with blindness follows a closely aligned pattern to that of their sighted peers, indicating that much of the learning can occur alongside sighted peers in mainstream classrooms.

However, participants also noted specific challenges for students with blindness at level 0, the visualization level. While the Van Hiele theory consists of five levels, participants’ statements primarily focused on level 0, suggesting that this stage requires particular attention for students with blindness. This finding is in line with Argyropoulos (2002), whose experimental study with students with blindness identified that while many struggled to reach level 1, some faced challenges even in achieving level 0. Although the methodologies differ, both studies highlight the significant difficulties that students with blindness encounter at the visualization level. In contrast, research on sighted students has shown that while many struggles to progress beyond level 3 or are in a transitional stage between Levels 0 and 1, they generally do not face challenges in reaching level 0 itself (Ma et al., 2015; Škrbec & Čadež, 2015; Trimurtini et al., 2022). Thus, this study further underscores that students with blindness encounter unique difficulties that sighted students do not.

Furthermore, participants identified bimanual exploration, hand coordination, and cognitive integration as important skills for students with blindness to reach level 0. As these skills are specific to those who rely on tactile perception, this highlights the need for differentiated teaching that focuses on developing these skills. This underscores the importance of implementing differentiated teaching which addresses these particular skills, early, as reaching the level 0 is a gateway for learning basic and more advanced geometry alongside the sighted peers in the mainstream school settings.

Finally, the choice between 2D and 3D representations was identified as an important factor in supporting students with blindness in reaching the visualization level. Tools that are too large or overly simplified 2D representations may place excessive cognitive demands on students, potentially hindering their progress. This underscores the importance of not only implementing differentiated teaching early in students’ schooling but also carefully selecting learning materials and adopting teaching strategies that are tailored to the unique characteristics of tactile perception.

Although the skills highlighted above have not previously been discussed in direct connection with geometry learning using the Van Hiele theory, past research emphasized their significance for students with blindness. For instance, Kershman (1977) identified subskills necessary for tactile discrimination, including active touch, fine hand movements, and whole-hand explorations. Additionally, in research on haptic perception, Lederman and Klatzky (2009) defined how haptic perception relies on multiple sensory inputs—such as cutaneous and kinesthetic input—along with working memory to retain images formed through tactile exploration. For example, when geometric figures are small enough to fit under a fingertip, they can be identified primarily through kinesthetic input. However, larger objects requiring the use of both hands (palms and fingers) demand additional reliance on cutaneous input and working memory, making the visualization process more complex.

From the above findings, it is evident that students who rely on haptic perception encounter unique challenges in reaching the visualization level of the Van Hiele theory, challenges that sighted students do not face. Several factors specific to students with blindness significantly impact their ability to “visualize” geometric shapes due to the nature of haptic perception, which mainstream teachers should be aware of. The role of specialists, such as teachers of students with visual impairments who understands the principles of haptic perception and can select appropriate learning tools is critical. Teachers with this specialized knowledge should collaborate closely with mainstream teachers and teaching assistants who may directly be involved in supporting children with blindness in the classroom to ensure students of this population can fully access and receive meaningful geometry instruction in mainstream classrooms.

Lastly, this research is not without limitations. As an exploratory study, it relied primarily on insights from a specific group of individuals in Japan. Future research should include a more diverse range of participants from various regions and cultures to enhance the broader applicability of these findings.

5. Conclusions

This study highlights that students with blindness encounter unique challenges that sighted students do not. To address these challenges, the study proposes effectively using differentiated teaching focusing on specific skills needed to reach level 0, such as bimanual exploration, hand coordination, and cognitive integration. Furthermore, special attention should be given to both the content of instruction and the learning tools used, ensuring they align with the characteristics of haptic perception. Since level 0 serves as a gateway to both basic and advanced geometry, it is crucial that differentiated teaching is provided early in students’ school lives. This support should involve collaboration with specialists in the education of students with visual impairments, who can guide the selection of appropriate tools to facilitate skill development.