Students with Visual Impairments’ Comprehension of Visual and Algebraic Representations, Relations and Correspondence

Abstract

Exploring learning trajectories based on student thinking is needed to develop the teaching curricula, practices and educational support materials in mathematics for students with visual impairments. Hence, this study aims to reveal student thinking through various instructional tasks and tactile materials to explore the sequence of goals in the learning trajectory. A teaching experiment involving introductory information on algebraic and visual representations regarding advanced mathematical concepts was designed for correspondence and relations. The research was carried out with a braille-literate 10th-grade high school student with a congenital visual impairment where colour and light are not perceived in Türkiye. As a result of the teaching experiment, the participant was able to determine the correspondence and relations between two sets using different representations. He even designed graphic representations using the needle page. The learning trajectory goals and instructional tasks can serve as guides for research on curriculum development, practice design and material development.

1. Introduction

The tendency in the education system is to place students with special educational needs in inclusive environments and to provide equal opportunities with their peers. However, as students with visual impairments (SVIs) are unable to observe visual elements such as symbols and shapes, their learning experiences may have deficiencies and difficulties (Aktaş & Argün, 2021; Şafak, 2005). These situations emerge in particular while developing concepts (Carney et al., 2003). However, curricula and universal standards focus on the contributions that mathematical knowledge and thinking skills have in daily life functions (Ministry of National Education [MoNE], 2018; National Council of Teachers of Mathematics, 2000). Therefore, the contrary to the view that SVs cannot perceive visual images, algebra is a remarkable research area for SVIs, as they have astonishing perceptual abilities (Millar, 1985). However, the lack of research in this framework (Cowan, 2011) points to the need for studies on designing instructional environments for SVIs (Buhagiar & Tanti, 2013). Therefore, research on the learning trajectories compose the basic theoretical framework for designing curricula (Clements et al., 2004, 2020; Simon, 2006) and learning processes (Battista, 2004).

1.1. Learning Trajectory

The hypothetical learning trajectory (HLT; see Simon, 1995) and learning trajectory (LT; see Clements & Sarama, 2004; Clements et al., 2020; Confrey et al., 2008) basically organise learners’ thinking and learning. These theories agree that LTs have the potential to regulate classroom practices. However, all students follow individual instructional sequences rather than a single instructional sequence in classroom practices based on LT. These individual instructional sequences are a step-by-step tracing of how student thinking develops for a particular concept (Confrey et al., 2008; Simon, 2006). Simon (1995) defined the HLT terminology based on the idea that every student experiences thought processes and impressions through the tasks the teacher prepares. Therefore, LT is defined as a map of how mathematical concepts and skills develop and describes how learners think regarding mathematical concepts.

Despite the absence of direct observation of learners’ comprehension and cognitive processes, learning roadmaps facilitate interpretation by delineating observable key objectives, skills and behaviours. As posited by Barrett et al. (2012), learning roadmaps provide an effective framework for informing teachers about the curriculum at every level and selecting activities. Similarly, Battista (2004) contends that they facilitate the evaluation of goals and the learning process. Despite the heterogeneity of the theoretical approaches proposed, researchers concur that learning roadmaps can be obtained through activities based on predictive teaching sequences (Simon, 1995), interactions in classroom applications (Steffe & Thompson, 2000) and concept-focused teaching sequences (Clements & Sarama, 2004). It is therefore possible to define a learning pathway map as the mapping of the development processes of a specific mathematical concept and skills, as well as the description of the learner’s thoughts regarding this mathematical concept. Furthermore, this process entails the identification of a series of tasks, i.e., a sequence of activities or objectives, in the planning of the learner’s prior knowledge and the learning processes related to the intended concept (Simon, 2017; Simon & Tzur, 2004).

Simon (2017) emphasised the need in the LT design to determine a sequence of tasks or goals based on abstraction. This task sequence requires the instructor to teach without giving hints or suggestions. Therefore, when designing an HLT, the concepts, learning environment, possible student misconceptions and student readiness should be considered (Simon, 1995). Perceptual diversity should be added to the criteria for SVIs due to their lack of vision (Millar, 1985). Hence, SVI’s comprehension and thinking processes can be obtained with the support of accessible materials (Aktaş, 2024; Bülbül et al., 2012). Thus, not only can the goals of LTs be obtained but also hints at how to develop and use material.

1.2. Students with Visual Impairments

In educational contexts, visual impairment can be categorised according to the extent to which the sense of sight can be utilised. Individuals with a visual acuity of less than one-tenth in both eyes, despite all available corrective measures, and who are unable to use their vision effectively for educational purposes, are classified as blind. SVIs, characterised by a visual acuity that falls between one-tenth and one-third of normal vision, require assistive devices such as magnifying glasses or corrective lenses to participate in educational activities independently (Şafak, 2005). The development of an individualised educational programme tailored to the specific nature of the impairment is essential for both groups.

Accessible teaching practices for SVIs seek to compensate for the limitations of sight by enhancing the use of other senses, particularly hearing and touch. Research by Goldreich and Kanics (2003) indicates that SVIs frequently exhibit heightened auditory and tactile perception in comparison to their sighted peers. These senses, in conjunction with verbal descriptions, serve as pivotal resources for the acquisition of concepts during cognitive development. However, Groenveld (1993) has noted that concept learning based solely on the descriptions of sighted individuals can restrict the depth and accuracy of understanding. Consequently, instructional sequences are recommended to commence with tactile approaches, subsequently transitioning to combined tactile and auditory methods. Nevertheless, it is important to note that certain concepts may persist in their abstract state, irrespective of the scale or magnitude with which they are presented. Mathematical language is characterised by its embodiment of concepts that are both highly visual and abstract, including algebraic notation, geometric figures and graphical representations (Edwards et al., 1995). Therefore, understanding how learning processes and cognitive development unfold in individuals with advanced tactile perception is of notable importance (Stevens et al., 1997).

As Groenveld (1993) emphasises, SVIs may encounter challenges in accessing learning experiences or comprehending visual elements, including objects, symbols, shapes and spatial relationships, that they cannot observe directly. Furthermore, empirical evidence suggests that increased severity of visual impairment is associated with lower academic achievement, which, in turn, negatively impacts the learning of mathematical concepts (Zebehazy et al., 2012). However, research has demonstrated that instructional practices and materials designed within an individualised and differentiated framework have the capacity to accelerate learning, enhance motivation and strengthen conceptual understanding (Agrawal, 2004). In this regard, the present study offers valuable insights for curriculum developers and educators working to optimise educational outcomes for students with visual impairments.

1.3. Algebraic Correspondence and Representations

Simon and Tzur (2004) emphasised gaps to still exist in the literature on teaching and learning for many concepts. According to Daro et al. (2011), algebra is the learning domain in which many of these concepts occur. Because algebra is a prerequisite for algebraic thinking skills, it is a tool for finding solutions to the difficulties encountered in daily life (Hunter & Miller, 2022; Kieran, 1992). For this reason, students should be able to understand and use basic concepts and symbols in algebra. For example, algebraic correspondence using set representations can be more useful in daily life for representing the concept of functions, which are defined as the mapping of objects and numbers (Aktaş & Argün, 2021; Kabael, 2017; Mosvold, 2008). Therefore, the importance of skills emerges for determining the relationship between the elements of sets, for using representation types and for translation between these.

It is a widely acknowledged fact that SVIs often experience difficulties in comprehending concepts that involve visual representations, such as shapes, pictures and graphs (Aktaş & Argün, 2021; Edwards et al., 1995). Nevertheless, it is evident that the fundamental concepts of mathematics are predicated on visual elements, including shapes, notations and graphs. In particular, graphs serve as vital instruments for the conveyance of information and the interpretation of data. It is widely acknowledged that graphs are of fundamental importance in the teaching of key mathematical concepts at all levels of education. These concepts include number lines, complex functions, curves and surfaces. Nevertheless, given their limited access to graphical representations, it could be argued that SVIs are at a disadvantage in this regard (Zebehazy et al., 2012). Accordingly, the research problem is ‘What are the LT and algebraic thinking processes of an SVI regarding the concepts of correspondence and representational types?’

2. Method

The study’s research design is a teaching experiment as the research data have been collected from individual instruction sessions and the participant’s cognitive structures have been investigated based on a mathematical concept (Cobb & Steffe, 1983). The main purpose of the teaching experiment is for the researcher as a teacher to understand or discover the student’s reasoning and thinking skills (Steffe & Thompson, 2000). For the teaching experiment, this study conducts clinical interviews during the instruction sessions using the interview protocols designed regarding the concept of correspondences between the elements of sets. Firstly, a pre-interview was conducted to determine the SVI’s mathematical knowledge and which tools to use in the instruction sessions (Steffe & Thompson, 2000). The interview protocols have been supported with concrete materials (e.g., braille printouts, needle page, 3D pen) and designed based on the findings obtained from the pre-interview. Also, the research was conducted in Türkiye.

2.1. Data Collection Tools

The data collection tools of the research are (i) the pre-interview, (ii) the clinical interview protocols/student tasks and (iii) the weekly clinical interviews, instructional sessions and video records. The clinical interviews were conducted to gather in-depth information and to examine the student’s cognitive development (Ginsburg, 1981). The participant’s mathematical learning processes are more important than his right or wrong answers in the clinical interviews. Therefore, instructional tasks were included in the clinical interviews. More decisive data were obtained through exploratory questions, successively related problems, and retrospective questions. As such, questions such as ‘Can you explain your solution?’ or ‘How did you decide?’ were included in the clinical interviews. The pre-interviews were conducted to determine the participant’s pre-knowledge regarding the concepts of correspondence and representation types as well as his demographic characteristics (e.g., his level of visual impairment, needs in the educational environment, educational experiences, braille literacy, using materials). Indeed, the learner’s initial knowledge is important for revealing the HLT (Simon, 1995; Simon & Tzur, 2004).

The hierarchical structure of the concepts and the curriculum (Ministry of National Education [MoNE], 2018) were considered when specifying the goals for the interview protocols. The literature was examined for concept definitions, historical development, student thinking, misconceptions, LT examples and suggestions for SVIs (Aktaş, 2020; Aktaş & Argün, 2020a, 2020b; Aktaş, 2022; Arieli-Attali et al., 2012; Bülbül et al., 2012; Cansu, 2014; Clements & Sarama, 2004; Cowan, 2011; Friedlander & Tabach, 2001; Horzum, 2016; Kabael, 2017; Kieran, 1992; Maulana, 2019; Moss et al., 2019; Panorkou et al., 2013; Şafak, 2005; Weber & Thompson, 2014). Next, the tasks were grouped according to context, and the interview protocols were designed for the five instructional sessions. The braille slate and stylus were used to take notes, and the cubarithm slate was used for mathematical operations and numerical notes. The needle page (Bülbül et al., 2012) obtained from the MoNE Course Tools Centre was used for the graphics. The longest stick and abacus bead were considered to represent the line and ordered pairs, respectively. Thus, a tactile material was obtained for designing graphics using string, electrical wire and rubber bands. The program TactileView and a 3D pen were used for tables and Venn diagrams (see Figure 1).

2.2. The Participant

SVIs may have different characteristics depending on variables such as the degree of their disability, visual experience, and presence of another disability. SVIs may also have learned to use their other senses to varying degrees and frequencies (Edwards & Stevens, 1994). Zorluoğlu et al. (2016) indicated blind individuals to have a more limited memory for learning and recalling concepts compared to individuals with low vision. Warren (1994) emphasised that, instead of comparing these groups, one group should be examined holistically and investigated in detail. As such, this study aims to determine the LT for an SVI with a high-order congenital visual impairment. This participant is referred to with the codename of Mete and was selected according to the criterion sampling method. The criteria are as follows: having 98% vision loss according to MoNE’s Regulation for Schools of Special Education and Rehabilitation Centre, having a congenital visual impairment, being unable to perceive colour and light, having braille literacy, being unable to use Latin letters (even with large fonts) and having no additional disabilities. Mete completed middle and elementary school for students with visual impairments and is a 10th-grader in a mainstream high school.

The data of the study were obtained with the approval of the ethics committee and participant consent with the report [Gazi University Ethics Committee, the report numbered E-42944]. Participants were volunteers and consent was obtained from the participants and their parents.

2.3. Data Analysis

On-going and retrospective analyses are used for the teaching experiment. The on-going analysis is an analysis method performed between teaching sessions that focuses on evaluating the results of the previous session and on designing the next session. Retrospective analysis is performed following the data collection phases and focuses on analysing the instructional sessions (Steffe & Thompson, 2000). The participant’s achievement of goals and progress along the LT have been determined by considering the participant’s discourse, hand gestures, ability to use materials, and thoughts regarding the concepts using these analysis methods. For example, the participant was able to show the one-to-one correspondence between two sets using an electrical wire and rubber band (see Figure 2) and searched for the symbol representing correspondence representation in braille. Thus, the clues obtained for the translation between the different representation types of correspondence were united, and the results were determined over a broad framework using retrospective analysis.

2.4. Research Process

In the teaching experiment process, ongoing analyses performed after each instruction session, received expert opinions, necessary revisions were made for the next session’s protocol, and pilot studies were carried out (see Figure 3). This process was repeated until the participant’s LT has been determined for the concepts; then the LT was obtained through the retrospective analysis. The individual clinical interviews were conducted with a participant codenamed Faruk for the pilot studies. Faruk has a similar background as Mete: He was born blind, has no perception of colour and light and is a 10th-grade student at a mainstreaming school. The pilot studies provided the researcher with the opportunity to gain experience in conducting the teaching experiment and identifying the need for inclusive practices. The realisation of the cycle and the progress of LT are discussed in the results section.

Opinions were requested from academicians with expertise in mathematics, mathematics education and special education regarding contexts such as determining the goals, the number of sessions, the accessibility of the examples, and tools for SVIs in regard to designing the interview protocols. Accordingly, revisions were made in terms of selecting tasks and materials and sequencing and increasing goals.

2.5. Validity and Reliability

The strategies of long-time interaction, continuous observation, experts’ review and participant validation were used to increase the trustworthiness of the research. Before the research, interviews were conducted with the participant and his family about the research content and participation approval for strengthening communication. The participant was given time to examine the office where the interviews were conducted and was informed about the location and angle of the camera. During the sessions, the participant was allowed breaks to rest. The transferability of the research results has been ensured by using the criterion sampling method and providing detailed descriptions. The reliability of the research has been increased by using experts’ opinions regarding the data collection tools, conducting pilot studies and having two encoders for the analysis.

3. Result

3.1. The Pre-Interview Session

Mete explained the set as ‘being a community formed by the coming together of elements’ and knew different types of representations. He stated that ‘function had been explained in class, but not with me’ in regard to correspondence between two sets. For the example of bus seats and seat numbers, Mete was able to express the elements of the sets to be about mapping. He was also able to explain and represent the axes, ordered pairs and the origin for the coordinate system on the needle page. However, he stated that he had never examined tables and graphics before.

3.2. Teaching Experiment Sessions

This section presents the results regarding the correspondence between sets, the relation for correspondence and the types of representation types.

3.3. Determining Correspondence Based on Relation

The session started with the example of mapping each integer to its square in order to discover the relation of correspondence between the elements of two sets. Mete explained this with the example, ‘I choose 17. 17 times 17 is 289, I write it and then multiply it by itself’. When asked to show this correspondence using a tool, Mete chose the braille slate. While Mete was mapping the elements, he said, ‘17 equals? Anyway, it could be wrong. […] I wrote 17 and the equals sign, and 17² and again the equals sign then 289’. He then searched for the symbol for correspondence. He wrote the Z and K sets, which he had determined according to the correspondence using the listing method with the braille slate and writing an arrow between the sets with the braille codes of 2-5 and 1-3-5. He tried to map the elements by embossing the line segments (see Figure 4).

Mete could not express a one-to-one correspondence when discussing how no other building could have the same house number as his house number for the houses on his street. For this reason, the discussion continued with the example of school numbers, and this time Mete noticed a one-to-one correspondence to exist between the sets of school numbers and students.

Mete was asked to place 9 TV channels however he wanted on the TV remote for the different representations of one-to-one correspondence. Mete wrote the set of TV channels and the set of numbers using the listing method and reasoned ‘Should I match directly since they have the same number of elements?’ for the first elements. Mete embossed the line segments for a few mappings, as these are easy to feel tactilely. However, turning the slate on and off each time by specifying the elements made mapping difficult. Therefore, he wanted to use rubber bands of different lengths to show the mappings and decided to randomly map the elements. He also showed this correspondence using the Venn diagram and explained the one-to-one correspondence as ‘an element can only be mapped to one element from the other set.’ Next, the change in the number of unemployed populations of a country by year was discussed as an example of using different representations. Mete noted the years on the left side and the number of unemployed people on the right side of the cubarithm slate with a space between them. While taking notes, Mete noticed the relation and thought aloud, ‘It increased by two hundred and the other by three’. The sets were represented with a Venn diagram using electrical wire, but the relation could not be represented algebraically (see Figure 5).

In order to increase Mete’s skill in translation between representations, the interview was continued with a cloud seeding scenario with a table. He determined the relation between the precipitation amount and the number of elements from the seeding number set, but he could not determine the relation between these sets. In addition, because Mete had compared the correspondence which he’d represented with the cubarithm slate, to the table for the previous example, the session continued with examples of table representations. Mete then said that the presented example of the baggage weight fees table resembled the cargo he had sent to his sister. While Mete was examining the top row of the table with one hand, he examined the bottom row with the other hand and noticed that some cargoes of different weights were mapped with the same price. He interpreted this situation as having ‘no standard’ and determined no one-to-one correspondence to be present. Thus, comprehension about the concepts of table, correspondence, and one-to-one correspondence were seen to have been obtained.

3.4. Examining the Relation Between the Elements of Two Sets Using the Coordinate System

When the ordered pair was first questioned for the representation of the relationship with the coordinate system, Mete stated that the ordered pair was represented by a point and showed the correspondence between the two sets. Thus, the session continued with examples of the correspondence presented using different representation types and the coordinate system.

While listening to the scenario about hourly variations of bicycle tour fees, Mete determined the sets and their elements and represented the correspondence with a table using the braille slate and stylus. Then, while representing the correspondence with the coordinate system using the needle page, he had to determine the distance between the two needles as 5 units on the axis of the wage set because of a limitation of the material. Mete emphasised that the units on the axes should be equal in the coordinate system and marked the points representing the correspondences without any difficulty, because Mete had previously learned how to construct the coordinate system with the needle page. He first determined the point on the x-axis with his right hand, then the point on the y-axis with his left hand, marking the point where his fingers intersected with a bead and intuitively determining the matches over time.

Mete interpreted the table about the calories an athlete burns per hour as ‘Here is 300 calories per hour’. The relation was queried for a different point compared to the points marked with the information on the table for representing with the coordinate system. So, when asked about the calories burned half an hour after starting the training, he interpreted it as ‘300 for 1 h, and 150 for a half hour’. He stated, ‘the matches are correct’ after placing a few beads at half hour intervals. He created the graphic with line segments by combining the points he had marked using the rubber bands (see Figure 6).

Mete stated Venn diagrams and table representations can be used for ordered pairs in the question, saying ‘according to the pairs, those on the x-axis to one side, those on the y-axis to one side. So, 2, 1, −1, −2, −3 are the elements of one set and 4, 2, 1, 0, −1 are the elements of the other set’. Thus, Mete had noticed the set of ordered pairs representing the correspondences between the two sets, but he had to think a little while marking the point (2, 0) using the graphic representation. When asked about the necessity of combining the marked points for graphic representation of the correspondence, he replied, ‘We don’t know the other points yet.’ Thus, Mete was determined to be unable to distinguish the difference between dot and line graphs based on the relation.

Mete followed the line segments with both hands while examining the correspondence with the Venn diagram for the dot graph (see Figure 7). He said, ‘since it is a line segment, I will continue straight’ for the intersections of the line segments and determined the correspondence. He noticed that two elements correspond to the same element and was able to represent the graphic using the needle page.

While examining the line graph at the unmarked points for the next example, Mete connected the marked points with a string (see Figure 8). He realised not all points should be combined when considering a new point in terms of the relation that creates the correspondence.

Mete determined the correspondence for the table representation of water pressure by distance under the sea, but he did not examine the relation between the elements of the sets. His misconception about the graphic representation was identified with the statement, ‘it would be a line… because it increases irregularly’. Thus, he was asked to determine the relation of the (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4) and (3, 9) correspondence. He said, ‘Is x squared y? Huh, y mapped to x squared […] I know the relation very clearly’ and designed the line graph. Thus, he distinguished the difference between the dot and line graph. Mete followed the graphic using rubber bands with his fingers and said ‘it looks V-shaped’. When the graphic was re-represented using the electrical wire, he described it saying, ‘Yes, this time it is U-shaped’ (see Figure 9).

The next session presented the graphic for y = x + 2 using string with some points marked using the needle page. Mete stated the relation as, ‘shifted by one on each axis, and shows the relation as a pattern […] for example (2, 4), (3, 5) hmm two more! Correspond to 2 add to the elements of the x-axes’. Thus, he explained the generalisation verbally while using the pattern and counting the needles, he marked the points (4, 6) and (−3, 1) before using the algebraic relation he had determined. He had difficulty with algebraic representation and said, ‘Is it x + 2 + y or 2y + 2?’. He examined the point (3, 5) as a different point but could not represent the relation algebraically.

Mete determined the points with the help of needles and examined y = x² − 4 as a parabolic graph using the needle page and marking some points. Because he focused on the relational increase or decrease, he could not determine the algebraic representation and said, ‘The graphic is non-linear’. Next, the ordered pairs according to y = x − 1 were given, but Mete felt it to only be a one-line graph and could not define the algebraic representation. Because Mete had difficulty shaping the electrical wire, he represented the graphic with the string.

Mete was asked to determine the relation represented by marking the ordered pairs as (0, 0), (−1, −1), (1, −1), (−2, 4), (2, −4), (−3, −9), (3, 9) on the needle page and to plot the graphic for the sub-sets of real numbers. Mete first focused on the relational increases or decreases, but when these relations were incorrect for each ordered pair, he noticed the square relation and was able to express the algebraic representation. Mete drew the graphic using string and showed how the graphic would continue by extending the string past the edge of the needle page (see Figure 10).

3.5. The Algebraic Representation of the Relation Between Two Sets

An example about the daily food intake for a blue whale was considered for the algebraic representation of the relation between the elements of sets. Mete determined the variable x to represent the elements of a set in the table, and while examining the second row, he said, ‘2 times 4 tons is 4x tons.’ Mete determined the variable y for the food intake set and reached the conclusion ‘x times 4y? […] for x days, 4 times x. Haa, y is equal to that. The food changes according to the number of days; y changes with respect to x.’. Mete was then able to determine the relation and variables between the elements of the sets in the example of the table showing the total distance travelled per hour. However, he said, ‘100 for 2, 150 for 3. 50x? 50y = x?’ and was unable to define the relation using algebraic representation. When Mete checked his answer for the (1, 50) correspondence, he gave the correct answer. Then, two sets were given using Venn diagrams, and their elements were corresponded based on the y = x − 3 relation. Mete said, ‘I guess it decreases by 3’ for the ordered pairs he had determined, but he considered the algebraic representation to be ‘−3x’. When he checked this idea for an ordered pair, he determined the algebraic representation as ‘x − 3’. Examples of algebraic representation were examined for the graphics y = x³ and y = x, respectively, and generalisations were checked for different ordered pairs. Thus, Mete realised that a generalisation cannot be expressed by examining an ordered pair.

3.6. Mete’s LT

Mete’s process for discovering the correspondence and relations between the elements of the two sets and using different representations following the goals in

Table 1. Mete’s LT.

Instructional Goals	Instruction Session
Correspondence between the elements of two sets	Examining the daily life examples for the ordered pairs, correspondence representations and correspondence with braille
Expressing the relationship between the identity elements of the set	Expressing the relationship between the identity elements of the sets using different representations
Exploring the correspondence and relation for the table representation	Determining the relation for correspondence using the table representation
Determination the relation between two sets	Examining the relation for the correspondence between the elements of two sets using different types of representations apart from algebraic representation
Mapping according to the relation between the elements of two sets	Determining the relation between the elements of two sets by using different representations and mapping the elements
Mapping one-to-one between the elements of two sets	Examining the one-to-one correspondence with different representations
Ability to identify points that represent correspondence according to relation on axes	Ability to represent the correspondence of elements according to the relation between two sets with ordered pairs using the needle page
Expressing the relation between two sets	Expressing the relation between two sets using different representations apart from algebraic representation
Ability to represent the relation with a graphic using points and line segments	Designing or examining the examples of dot and line graphs using the needle page. Initially examining the linear relations and discussing how to connect the points marked on the line graph
Ability to represent the relation between two sets using a graphic	Ability to graphically represent the relation between two sets presented with different representation types. Ability to identify points on a graphic
Ability to express the relationship between two sets represented with a graphic and by different representations	Expressing the relation between two sets represented graphically with different representations based on the critical points
Ability to express the relation between two sets with different representations	Representing the relation between two sets using the table, Venn diagram, ordered pairs, graphic and algebraic representations using braille, the cubarithm slate, or needle page and translating between representations

.

4. Conclusions

The findings of this study demonstrate that Mete was capable of developing a meaningful understanding of the correspondence between sets through tactile and multi-representational learning experiences. Throughout the teaching experiment, Mete demonstrated progress in identifying and expressing relations using various representational forms, including braille slates, cubarithm slates, Venn diagrams, tables and coordinate systems. Despite initial difficulties, particularly in the domain of expressing one-to-one correspondences and algebraic representations, he demonstrated a gradual improvement in his ability to generalise relations and interpret diverse forms of representation.

The utilisation of tactile materials, including needle pages, rubber bands and string, served to enhance Mete’s comprehension of mathematical correspondences in terms of both space and concept. His engagement with concrete and symbolic representations enabled him to intuitively construct mappings, compare representations and detect patterns. It is noteworthy that Mete’s comprehension advanced from a fundamental awareness of sets and mappings to a more sophisticated discernment of functional relationships and graphical representations, encompassing parabolic and linear forms.

Despite persistent challenges in algebraic generalisation, particularly during the transition from visual or tabular data, Mete demonstrated an enhancement in the verbalisation of patterns and the testing of relational hypotheses. The findings of this study indicate that multisensory and scenario-based teaching strategies are of pivotal significance in facilitating the comprehension of abstract mathematical concepts, such as function, mapping and algebraic reasoning among visually impaired students. To conclude, Mete’s learning trajectory highlights the potential of accessible, hands-on learning environments to enhance the conceptual development of mathematical ideas in SVIs.

5. Discussion

The LT obtained as a result of this research can be asserted to be generalisable for SVIs. However, the clues for the learning processes can be obtained for how SVIs’ comprehend the concepts of correspondence and relation by comparing the results with the literature (Arieli-Attali et al., 2012; Maulana, 2019; Moss et al., 2019; Panorkou et al., 2013; Weber & Thompson, 2014). Accordingly, the elements reflected in SVIs’ comprehension and LT can be related as in Figure 11. For example, SVIs who aim to use algebraic representations are likely to use materials and prefer different representation types due to the difficulties of braille.

Pre-knowledge noteworthily plays a role in LT (Clements et al., 2020; Simon, 1995; Simon & Tzur, 2004) regarding the goals about the algebraic representation of the relation between two sets. The inadequacies in Mete’s pre-knowledge of algebraic expressions played a role in ordering the goals and designing the instructional sessions. Thus, the goal of being able to express the relationship between two graphically presented sets using different representations was added to the LT.

The roles of supportive instructional tools and braille emerged in SVIs’ choice regarding the representation types for concepts. Indeed, Mete had difficulty with the correspondence representations due to the symbols on the cubarithm slate being insufficient (Aktaş, 2022; Maulana, 2019) for representation of the examples using the listing method. However, concrete materials such as the cubarithm slate are interestingly suitable for set representation with Venn diagrams. Emphasising to the SVI that the representations that use these concrete materials are Venn diagrams is sufficient enough. In addition, the uselessness of the braille slate as a listing method indicates the Venn diagram to be a more accessible representation for some materials, as mathematical representations in braille require many characters to be written (Aktaş, 2024; Edwards & Stevens, 1994).

Examples in the context of scenarios based on SVIs’ experiences (Mosvold, 2008) increase their success in interpreting graphics. Examining examples such as numbers that students are familiar with and then matching each integer with its square where algebraic representation is needed also play a role in students’ ability to comprehend the correspondence between two sets. SVIs may need to use symbols for the correspondence representations and their guesses. For example, Mete preferred the ‘=’ sign for showing the correspondence between the elements of sets because SVIs can operate by leaving a blank space instead of the equal sign on the cubarithm slate (Aktaş, 2022; Cansu, 2014). However, this choice of representation is known to create misconceptions in advanced concepts (Aktaş, 2020; Aktaş & Argün, 2020b). Mete’s choice for the concept of correspondence has added a new item to the list of these misconceptions.

Unsurprisingly, the materials used in instruction sessions affect the LT’s goals and student comprehension, because instead of the ‘bar’ material upon which Simon and Tzur (2004) had structured LT, concrete materials such as the needle page were used in the current research. The difficulties that these materials cause in the instructional sessions are not limitations of the research. Ultimately, these materials ensure the realisation of the LT’s goals. For example, the number of needles being insufficient for the number of elements of the set seemed to be a limitation, but this allowed the goals to be repeated regarding the correspondence between sets. Therefore, the designed axis that Friedlander and Tabach (2001) consider a disadvantage for visual representations reinforces the comprehension of graphic representation. Here, the extent to which the SVI had been engaged with the material is important. For example, the frequent use of the cubarithm slate (e.g., for taking notes) facilitates the comprehension of vertical tables in particular.

Contrary to the claim that symbolic representations are the most understandable for students (Friedlander & Tabach, 2001), SVIs would primarily prefer graphical representations for representing relations between clusters. These can be used especially for note-taking and developing practical strategies to mark ordered pairs on the coordinate system. Thus, the needle page can be highlighted as being an easy to use and tactile material for SIVs. Also, graphic representations are more practical for SVIs’ comprehension of graphics, correspondence and interpreting the relation. This result can be supported by Friedlander and Tabach’s (2001) interpretation that graphic representation provides a picture of the relation between the elements of sets quickly and efficiently. However, the preference for graphical representation may also be related to the lack of concept knowledge about algebraic representation. In this way, difficulties are experienced because relation that cannot be determined using algebraic representation can be easily determined using the needles. In addition, the difficulties regarding algebraic representation may create misconceptions about closed curves when designing a line graph. Moreover, discourse (Aktaş & Argün, 2020a) or linear relations (Aktaş, 2022; Arieli-Attali et al., 2012; Hunter & Miller, 2022) through the needles can be used to determine the pattern. Discussing the relation for missed points in particular where generalisations are made and examining the relation examples apart from linear relations are effective strategies for these cases. Therefore, LT should include goals that consider these strategies regarding the transition between algebraic and graphic representations similar to Weber and Thompson’s (2014) goals.

One instructional session may be insufficient for the goal of expressing the relation between two sets using algebraic representation due to the need to first discuss what should be understood from the concept of relation, followed by how to determine the relation between the elements and that this relation is not limited to just increasing or decreasing. However, despite determining the relation between two sets, difficulties may exist in the algebraic representation, or the relation of correspondence cannot be determined while focusing on the relation between the identity elements of the sets because Maulana (2019) emphasised SVIs’ individualised education programs to have not been designed in accordance with advanced algebraic concepts. For LTs (Aktaş & Argün, 2021; Clements et al., 2020; Moss et al., 2019; Panorkou et al., 2013) the concept of variable is additionally understood to have been examined with letter expressions or as variable representations.

6. Limitations and Further Research

The LT and instructional sessions that resulted from this study will enable curriculum development (Clements et al., 2004), the design of individualised education programs (Kurth et al., 2019), assessment tools (Battista, 2004; Suh et al., 2019) and supportive instructional tools, generally developing the learning standards (Simon, 1995; Simon & Tzur, 2004; Suh et al., 2019) for SVIs and designing research on them. LT is affected by the tasks presented in the instructional sessions and the supportive instructional tools (Simon, 1995). However, the results include the guiding clues that consider when conceptual understanding has been realised at the end of the instructional sessions. An important clue based on the concept is that the goals and tasks about daily life examples for the correspondence between two sets facilitates comprehension. The characters used in braille should be considered too numerous for SVIs to take notes (Aktaş & Argün, 2020b; Edwards & Stevens, 1994) and too difficult for showing correspondence for the examples of number sets. Therefore, including Venn diagrams or tables for the correspondence representation with concrete materials such as a cubarithm slate and needle page as well as the sets using the listing method for practice will simplify the opportunities.

SVIs’ familiarity with the tools supports their comprehension before the practices with a supportive instructional tool (Aktaş, 2024; Cowan, 2011). For example, SVIs can easily identify ordered pairs once their experience with the needle page increases. In addition, SVIs can use familiar tools for comprehending new concepts. For example, using the cubarithm slate for vertical table representation is interesting. Therefore, SVIs’ experience with the cubarithm slate may be increased first before examining examples on table representation and vertical tables when teaching tables. However, ideas that cause misconceptions should be avoided when using the cubarithm slate and needle page. As such, the learner should be prevented from using identity symbols or performing algebraic operations using the tools’ equipment.

The results of this research are limited to one congenital SVI’s LT that was obtained as a result of the instructional sessions designed with the determined supportive instructional tools. However, this limitation can be eliminated by future research carrying out similar studies and by enriched instructional practices. Pre-knowledge, experiences and tools cause differences in student thinking and LT’s goals (Simon, 1995, 2017). Therefore, LT’s goals and tasks should be arranged in this context and included in practices.