Counting Skills of Young Children With Visual Impairments

By Thuy Phan and Paula Conroy

Thuy Phan is a doctoral student at the University of Northern Colorado who specializes in teaching literacy and mathematical concepts to very young children with visual impairments.

Paula Conroy is a certified Teacher of Students of Visual Impairment and Orientation and Mobility Instructor. She is a professor at the University of Northern Colorado in the School of Special Education who coordinates the Visual Impairment and Orientation & Mobility Programs. 

Abstract

The effective teaching of counting skills requires teachers to understand the fundamental concepts related to the development of basic math skills—what they are and how they develop. Teachers of Children with Visual Impairments (TVIs) need to use evidence-based strategies to teach counting skills to help their students understand number and quantity. This article explains the principles of the development of counting skills and typical difficulties children with visual impairments may experience. It provides several strategies that teachers can use to support children with visual impairments in developing counting skills. A case study illustrating the application of these evidence-based teaching strategies is presented.

Keywords

Counting skills, teaching strategies, children with visual impairments

 

Counting is fundamental to building mathematical skills, but it is not a simple concept. Children need to understand the nature of the different counting principles to master counting skills. According to Gelman and Gallistel (1978), the five counting principles are: 

1. One-to-One Correspondence: When counting, students assign one (and only one) number to each object of a set. Partitioning (separating) and tagging help with this skill. In one-to-one correspondence, it is crucial to get the correct partition–touch or move one object, tag it, then assign a number to that one object. The purpose of this principle is to compare the quantity of two sets and follow the steps of making the items in the first collection in a line. Then each item in the second collection is assigned with an item of the first collection. Finally, students need to identify the quantified relation of the two collections.
2. Stable Order: The nature of this principle is that children can count and say number names in a correct sequential order. 
3. Cardinality: According to Bostelmann (2007), the Cardinality Principle includes: (a) a number in sequence is assigned to an item; (b) the first number is always one, and the last number represents the total quantity of the collection (e.g., 1, 2, 3, the total number in the set is 3).
4. Abstraction: Understanding that how you count stays the same no matter what you count. For example, any set of objects can be counted as a set, regardless of whether they are the same color, shape, size, etc. This can also include non-physical things, such as sounds, imaginary objects, etc.
5. Order Irrelevance: Knowledge that the order that items are counted in (e.g., left to right, right to left, in a random fashion) is irrelevant, as long as every object in the set is given only one count.

These five counting principles establish the developmental progression of children in learning number sense that is crucial for teachers to understand so they can apply appropriate instruction for their students. Interpreting the nature of one-to-one correspondence and cardinality principles helps children learn abstraction and order irrelevance in counting skills. Further, order irrelevance is a concept that develops as children work on their one-to-one correspondence skills, especially when teachers are intentional and explicit in their teaching. It is critically important for teachers and parents to create opportunities for young children to experience the above counting principles, as these experiences will facilitate the understanding that is needed to master counting skills.

Many children, including those with visual impairments, do not understand the fundamental nature of counting because they have not been explicitly taught these five principles. Counting skills are fundamental to mathematical knowledge and so are essential for moving to higher-level mathematical concepts (Jacobi-Vessels et al., 2016; Lee et al., 2012; VanDerHeyden et al., 2006). Research shows that if children do not understand the nature of counting principles, they will have difficulty learning math concepts, which affects long-term achievement in mathematics (Aragón-Mendizábal et al., 2017; Fuhs et al., 2016; McGuire et al., 2012; Perna & Loughan, 2014). This, along with the unique learning characteristics of children with visual impairments, makes the development of counting skills an essential topic for TVIs.

Learning Characteristics of Children With Visual Impairments

Various factors can impact how children learn number sense during the preschool years (Perna & Loughan, 2014).  The development of mathematical skills and knowledge in sighted children depends on their visuospatial working memory (Fanari et al., 2018), social-emotional issues (Moomaw et al., 2010), sociocultural context (Karsli et al., 2015), and the nature of the principles of counting skills (Lago & DiPerna, 2010). Like their sighted peers, children with visual impairments are strongly influenced by the above factors plus several others. Children with visual impairments have fewer opportunities to learn incidentally (Erickson et al., 2007; Mon & Yap, 2012; Smith & Smothers, 2012). Consequently, they may be passive in exploring mathematical learning materials and new materials. This lack of exploration may affect the development of mathematical concepts and skills.

The development of fine motor and hand manipulation skills may be delayed in children who are visually impaired (Erickson et al., 2007). These motor skills play a crucial role in manipulating objects when learning counting skills. Skills such as pointing, counting in a row, juxtaposing, and superimposing are just a few examples (Bostelmann; 2007). However, students with visual impairments can learn mathematics included in the general education core curriculum when provided appropriate instruction (Klingenberg et al., 2019). Interventions for the effective teaching of counting skills require teachers to understand the challenges children with visual impairments may experience in learning to count and find solutions because counting is such an important and foundational mathematical skill.

Difficulties Specific to Counting for the Visually Impaired

Children with visual impairments may experience several common problems developing counting skills. They may have limited conceptual understanding of counting due to lack of exposure or access. This may cause confusion in learning foundational mathematical terms of comparison like “many” and “few,” “same” and “different,” “large” and “small,” and “part” and “whole” (Smith & Smothers, 2012).

Omitting Objects When Counting

Children with VI may miss counting some objects due to the visual characteristics of the objects. It is challenging to count all of the elements in a set without seeing the whole set. They may guess the link between the number sequence and the number words without understanding the relationship between the number and the context.

Repeated Counting

Children with VI may count the same object repeatedly or might omit them altogether (Bostelmann, 2007). They may touch or repeatedly point to objects if the set of objects/items are not well organized or are moveable. For instance, when counting the total number of manipulatives, the student with VI may have difficulty locating all of the items, and if the items move during counting, the counter may lose track of them and count them twice.

Number Words That Do Not Follow the Correct Order

When children do not know the correct words for the numbers up to ten, they may count incorrectly, e.g., “one, two, four, five, seven." The names of the numbers need to be taught in the correct order as a part of the teaching process. This is frequently overlooked or assumed.

Problems With the Nature of the Cardinal Principle

Many children do not realize that the last number of the number sequence determines the quantity of a set. This demonstrates that the child is counting without awareness of cardinality. Other children may tag an object/item with more than one number word. They may also count by rote memory. As a result, even after counting, a child may not know the total number in a set.

Order Irrelevance

Children with visual impairments may become confused when attempting to identify the position of objects when counting. They may repeat count because the items are disorganized. Hence, children need to organize the whole set or use other counting principles. Children may have their own routines or strategies when counting, like always reading from left to right (as when reading prose) or counting by putting the object into a small container when it has received a number. Once the items to be counted are organized, the child can come to understand that the order in which they count the objects does not affect the quantity of objects in the set.

The difficulties mentioned above make it important to identify where the breakdown in counting skill development is occurring for the individual with a visual impairment. Solid counting skills help students gain mathematical knowledge, foster the capability to work with mathematical problems, and lay the basis for acquiring higher-order mathematical skills and concepts (Jacobi-Vessels et al., 2016; Lee et al., 2012). Thus, educators need to use appropriate strategies to teach the child with a visual impairment such vital skills.                                                                                                       

Strategies to Teach Counting Skills for Children With Visual Impairments

Using Accessible Materials and Assistive Technology to Teach Counting Skills

Adapting materials, using tools, and providing appropriate accommodations are essential for children with VI in learning (Smith & Smothers, 2012). Rottmann et al. (2020) reviewed the literature in this area and identified some important considerations for assessing materials for use with children with VI:

• Because students with visual impairments need more time to explore and become familiar with new materials, using unique items every day can detract from learning rather than enhance it. Teachers should use the same materials for a length of time and for many different tasks so that extended time for exploration is not needed for every lesson. Providing variety is essential for engagement and motivation, but introduction and use of new materials should be strategic.
• Clearly structure and emphasize all materials used, such as describing inherent structures to enhance concrete experiences.
• Openness–use various sensory channels for different mathematical content, documentation, and communication. Allow children to present their ideas through demonstrated use of materials, auditory response, or visual response (like fixation).
• Teachers can use visual materials and methods when teaching children with VI, but they need to be certain the student has access to the content. High contrast, magnification, and use of low-vision tools need to be allowed.
• Use of multisensory methods and math materials can create more opportunities for children to learn basic math concepts using hands-on activities in their surrounding environment. The use of real objects, 2D materials, accessible manipulatives, and assistive technology tools can help students with VI learn math concepts (Smith, 2017).

Assistive technology and accommodations can help the child develop mathematical concepts. Some of these include the use of concrete mathematics aids (manipulatives), applied finger-counting, talking calculators, the Math Window, the Brannan Cubarithm Slate and Cubes, the Cranmer Abacus, and a braille writer. Computer games and applications on an iPad/smartphone with pictures and auditory components are helpful learning resources for students with VI. Children with visual impairments can learn fundamental math skills, like counting, when they are provided appropriate instruction by teachers trained in visual impairment, accommodations, and in the use of tools or devices (Smith & Smothers, 2012). For example, teachers can use an abacus to teach one-to-one correspondence and place value, but they may hesitate to introduce the abacus to their students if they lack knowledge and skills of using the abacus themselves (Amato et al., 2013). Hence, to enhance the understanding of counting skills, teachers need to be trained in how to use the tools and in how to teach students to use the tools or devices for solving math problems.

Applying Finger-Counting

Finger-counting plays a crucial role in the early physical representation of a number (Lee et al., 2012). The finger-counting gesture can assist young children in various hands-on manipulations, such as pointing or touching in one-to-one correspondence or stable order. Finger-counting can also present a connection between the preverbal and verbal systems of numerosity (Andres et al., 2008; Fayol & Seron, 2005). Children count on their fingers and use finger-counting as a natural way to easily remember concepts. However, students who are blind may face difficulties in motor skills and eye-hand coordination, which leads to problems in using fingers to count correctly. Finger-counting strategies have been formalized and rely on recognition of specific hand movements (Crollen et al., 2011). Thus, it is necessary to teach children with VI how to use finger-counting properly in order for them to apply this strategy to enhance their learning. 

Using Specific Instruction

Using specific instruction is an appropriate approach for teaching all children, including those with VI. Klingenberg et al. (2019) claimed that tactile learners need additional time and direct instruction to learn specific skills. Tactile learners also need accurate information and detailed feedback about their task to access information and to practice skills. Ahlberg and Csocsán (1999) found that children with VI use their hands to locate the physical space of a set of objects. In this way, children can organize, learn the structure, and explore the wholeness of a set. Counting follows a line (or linear) order, so it may be the easiest way for children with VI to learn. Teachers may consider teaching children with VI using the following steps outlined by Kapperman et al. (1997) in Table 1.

Table 1
Linear Learning Steps

Steps	Guidance
Scanning	Move the hands across the top of each item in the array to be counted. The child can approach items and learn their characteristics.
Organizing	Organize all items to one side as a line if they are randomly displayed. When items are in a line, children can use their hands to locate the first items and explore the arrangement more systematically.
Partitioning	Options to count: -Count individual items and move counted items to a separate area on a tray or defined workspace. -Pick up items one at a time, give them a name, and place them apart from those yet to be counted. -Touch each item to be counted individually with one hand, giving each numeral name, while the other hand tracks the next item to be counted (Kapperman et al., 1997, p.16).

Using Modeling

Modeling is an approach that can be used to promote cognitive development of children with visual impairments (Ibrahim, 2015) and is an integral part of explicit teaching. Mononen and Aunio (2016) found that applying explicit teaching in counting skills instruction enhanced mathematical performance of children with low-performing math skills. There are several considerations in using modeling with young children with VI. In direct modeling, students are guided with verbal directions as the first step in using an explicit-teaching approach. The teacher and student follow an “I do, we do, you do” routine. Applying an explicit-teaching approach can scaffold a child’s understanding and then allow children to independently practice manipulating objects to learn early math skills. It is vital to model step-by-step (i.e., divide a task into small steps) using concise language with specific actions. In modeling with students with VI, teachers may use a hand-under-hand approach to guide each step. If a child has low vision, teachers need to make sure modeling takes place in the student's visual field and may need to use low-vision devices to make the activity accessible. Modeling should be used with appropriate descriptive language in demonstrations and instructions (Trương, 2001). Children may listen to an entire presentation first and then practice each step presented with the instructor. The final part of explicit teaching is to provide independent practice that leads to demonstration of understanding. The teacher decides the amount of information to present and the acceptable mastery level based on the child’s individual learning style and previous knowledge.

Using Appropriate Questions

Teachers should choose appropriate questions in teaching counting skills. Educators can use various types of questions to model counting skills and present learning materials (Trương, 2001). They can also ask questions about the name of the action, how the children can complete the action, how children feel about the action, the action's meaning, and the action’s results (Trương, 2001). For example, teachers can ask "what are you doing," "show me how you count," “do you think there is a lack of parking for this car," "why do you need to count the two rows," "are there more red cars than green cars,” “which set has more,” “which set has less,” and "why do you know the below row has more toys than the above row."  Asking questions should be based on the child’s abilities (e.g., copy characteristics of objects; who, what, where questions; questions about the relationship between the set or objects; questions about the whole process; creative questions; problem-solving questions; math reasoning).

Utilizing Sensory Integration to Teach Children With VI Counting Skills

Children with VI learn mathematical skills and experience through their senses (Ahlberg & Csocsán, 1999). Visual impairment impacts the way children learn because vision offers significant access to information to enhance the understanding of math concepts (Smith & Smothers, 2012). Still, children with visual impairments can use their remaining vision and other senses, such as tactile, haptic-tactile, and auditory, to explore the environment and perceive numbers.

Using Story-Based Math Problems

Integrating early numeracy skills, including total mathematical ability, quantitative capacity, one-to-one correspondence, and literacy or story-based math problems can enhance counting skills. Shared reading time with mathematical content can promote early numeracy skills in children with developmental delays (Green et al., 2018). Using playful math instruction could improve mathematical skills while children have a fun time playing to learn.

Using Meaningful Daily Activities and Play to Teach Counting Skills

Children enjoy learning through playing and daily activities. Children with VI can play various games to enhance their early math skills. For example, a board game or computer app can teach blind children about concepts of number and addition (Arvanitaki & Skoumpourdi, 2021). Stipek (2017) found that children are exposed to opportunities to learn math in many different settings throughout their day. The home environment is loaded with naturally occurring opportunities for parents to expose their children to early numeracy skills (Segers et al., 2015). Cameron (2020) found that using routines is a form of play as it supports students learning math through routines in the early childhood classroom. By participating in different, meaningful daily activities, children with visual impairments can practice their counting skills in their naturally occurring environment (Liedtke, 1998). When teachers facilitate counting opportunities in a meaningful environment for children, they are using a child-centered approach to learning math concepts because instruction is based on the individual child’s unique routine and environment (Parks et al., 2018).

Case Study Example

Daisy is a five-year-old child with blindness and a visual diagnosis of retinopathy of prematurity (ROP) along with additional disabilities. She has strong fine-motor skills and loves to pretend play. When Daisy was first evaluated, she did not demonstrate counting skills. She incorrectly responded when counting sets of up to five objects. When the teacher presented five toy ducks on a table and asked Daisy to count them, she repeatedly miscounted one object. Daisy also double-counted several items, giving them more than one number. She attempted to touch each object to count, but then counted one or two objects more than once. Daisy did not have difficulty ordering numbers and could rote count up to 20. Tea is a Teacher of Students with Visual Impairments, who is responsible for teaching Daisy math. By examining Daisy's learning characteristics and family history, Tea sought strategies to enhance her counting skills. After observation and discussion with Daisy’s classroom teacher, problems that Daisy had in learning counting skills were identified and teaching strategies along with interventions were implemented.

Several stages of intervention were implemented to teach Daisy counting skills. These were based on the TVI’s understanding of counting skills in both theory and practice. In each stage, the background of counting characteristics and strategies to teach counting skills for children with VI were applied.

Stage 1: Identifying Difficulties That a Child With VI May Face to Learn Counting Skills

To identify the problems that Daisy might have, Tea and the classroom teacher observed how Daisy performed counting tasks. The teachers also asked her to count in different orders and played various games involving counting skills. Then, they discussed her performance to pinpoint where the challenges were in Daisy's skill development. They recognized that Daisy omitted objects when counting and she often repeated counting. Daisy also had problems understanding the principle of cardinal counting.

Stage 2: Explain the Problems and Find Appropriate Learning Materials That Could Work for Daisy

One notable observation was that objects moved when Daisy was touching them during counting. This led to miscounting and repeated counting. Hence, it was recommended to secure objects in a stable line, so then the objects would not move while Daisy tried to touch them. It was important for the teachers to present the objects in the appropriate size for Daisy (i.e., not too big or too small) so she could use her remaining vision and tactual exploration skills. It was hypothesized that Daisy might not count repeatedly if all objects were put in a container or a box as she counted them. So, the teachers tried presenting the objects in a stable line initially and then Daisy moved each object to another container/box during or after counting. Tea accommodated the learning materials to teach counting skills based on Daisy's functional vision and skills. Tea added Velcro to objects, trays, and containers so that the objects or items could attach firmly and stay in one place. Contrasting colors and tactile items were used, as were an abacus and a large model of a braille cell.

Stage 3: Choose and Apply Appropriate Strategies

The teacher taught Daisy to use the two-hand approach described by Kapperman et al. (1997), as outlined in Table 2.

Table 2
Applying Kapperman’s Strategies for Teaching Daisy to Count

Steps	Guidance for Teaching Daisy Learning Counting
Scanning	Daisy was taught to use both hands crossed slightly on the top of a set to be counted. She also was given time to explore the items before she was expected to use them for math. Daisy missed counting some objects, so Tea reminded her to scan slowly and make sure she located all elements of a set.
Organizing	In the beginning, Tea let Daisy try counting the set independently, and then Tea edited her items to make sure they were all in a linear fashion. She also taught Daisy to use magnetic items, zippers, and an attachable foam line to make the object form a straight line with an appropriate distance for Daisy to organize.
Partitioning	Daisy was able to manipulate objects efficiently. However, Tea acknowledged that Daisy frequently stopped counting and became distracted by unfamiliar objects. Thus, integrating the counting skills with Daisy's favorite subjects, stories, and toys could help her maintain Daisy's attention and motivation to complete the requested counting task. Tea shared that boxes, containers, trays, and egg cartons were vital tools in supporting a child who is blind or visually impaired maintain organization of objects.

Daisy understood that each object should be tagged with a number word following a sequence. However, she typically stated an incorrect answer. Additionally, Daisy did not feel confident in reporting her answer. Tea decided that Daisy might not understand the principle of cardinality (i.e., the last number of the number sequence is to present for the total number of the set). Hence, Tea backed up and presented a set of just two objects. Once Daisy mastered reporting the set of two, Tea challenged her to count a set of up to five and then ten until Daisy mastered this principle.

Finger-counting with music and songs involving fingers worked for Daisy. The songs were used during break time in her class. Counting by fingers helped Daisy to remember the principles of cardinal counting. Questioning to check Daisy's knowledge and counting skills was necessary. For example, Daisy was asked “how many fingers,” “could you check how many fingers on your hands have toy rings,” “how do you know there were a total of six objects,” and “how did you use your hands to count.”

The multisensory activities that Daisy used included counting the number of clapping hands, stamping feet, and touching shoulders. When playing with sand, Daisy was asked to count the total number of sand cups that she found. While learning the music lessons, Daisy was asked to count the total number of musical instruments she heard.

Daisy loves playing and listening to stories and music. She is also very active and willing to explore new objects and toys. Therefore, after reading a short story that contained counting concepts, Daisy was asked to play games that required her to count. Daisy was also encouraged to count food items and steps at her school. She counted her movements during physical activities. Providing opportunities for children with VI to count throughout naturally occurring daily activities using everyday items, such as candy, coins, beads, Cheerios, and buttons is easy and provides ample practice opportunities throughout the day.

Using the above strategies, it only took Daisy one month to master cardinal counting skills. She had increased confidence to count correctly, and she could count accurately up to 20.

Conclusion

Counting skills are fundamental to the development of number sense. Understanding the importance of counting skills and the difficulties that children with visual impairments may have in learning them is important knowledge for teachers who work with this population. Teachers can enhance learning by identifying areas of difficulty in the development of counting skills and applying appropriate strategies presented here. Through assessment and evidence-based intervention strategies, children with visual impairments can make significant gains in their early number sense and problem-solving skills development (Root et al., 2020).

References

Ahlberg, A., & Csocsán, E. (1999). How children who are blind experience numbers. Journal of Visual Impairment & Blindness, 93(9), 549-560. https://doi.org/10.1177/0145482X9909300902

Amato, S., Hong, S., & Rosenblum, L. P. (2013). The abacus: Instruction by teachers of students with visual impairments. Journal of Visual Impairment & Blindness, 107(4), 262–272. https://doi.org/10.1177/0145482X1310700403

Andres, M., Di Luca, S., & Pesenti, M. (2008). Finger counting: The missing tool? Behavioral and Brain Sciences, 31(6), 642–643. https://doi.org/10.1017/S0140525X08005578

Aragón-Mendizábal, E., Aguilar-Villagrán, M., & Navarro-Guzmán, J., I., & Howell, R. (2017). La mejora del sentido de número en niños de preescolar con bajo rendimiento en matemáticas [Improving number sense in kindergarten children with low achievement in mathematics]. Anales De Psicología, 33(2), 311-318. https:// doi.org/10.6018/analesps.33.2.239391

o Arvanitaki, A., & Skoumpourdi, C. (2021). Adaption and evaluation of a learning-oriented board game for blind children. Journal of Blindness Innovation & Research, 11(1). https://dx.doi/10.5241/11-199. Retrieved from https://nfb.org/images/nfb/publications/jbir/jbir21/jbir110103.html.

Bostelmann, K. L. (2007). Teaching mathematics for all students at primary school and special primary school.  Hue University.

Cameron, A. (2020). Early childhood math routines: Empowering young minds to think. Stenhouse Publishers.

Crollen, V., Mahe, R., Collignon, O., & Seron, X. (2011). The role of vision in the development of finger-number interactions: Finger-counting and finger-montring in blind children. Journal of Experimental Child Psychology, 109(4), 525–539. https://doi.org/10.1016/j.jecp.2011.03.011

Erickson, K. A., Hatton, D., Roy, V., Fox, D., & Renne, D. (2007). Literacy in early intervention for children with visual impairments: Insights from individual cases. Journal of Visual Impairment & Blindness, 101(2), 80-95. https://doi.org/10.1177/0145482X0710100203

Fanari, R., Meloni, C., & Massidda, D. (2018). Visuospatial working memory and early math skills in first grade children.  In D. G. Sampson, D. Ifenthaler, & P. Isaias (Eds.), 15th International Conference on Cognition and Exploratory Learning in Digital Age (CELDA 2018) (pp. 127-133). International Association for Development of the Information Society (IADIS). https://files.eric.ed.gov/fulltext/ED600618.pdf

Fayol, M., & Seron, X. (2005). About numerical representations: Insights from neuropsychological, experimental, and developmental studies. In J. I. D. Campbell (Ed.), Handbook of Mathematical Cognition (pp. 3–22). Psychology Press.

Fuhs, M. W., Hornburg, C. B., & McNeil, N. M. (2016). Specific early number skills mediate the association between executive functioning skills and mathematics achievement. Developmental Psychology, 52(8), 1217-1235. https://doi.org/10.1037/dev0000145

Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Harvard University Press.

Green, K. B., Gallagher, P. A., & Hart, L. (2018). Integrating mathematics and children’s literature for young children with disabilities. Journal of Early Intervention, 40(1), 3-19. https://doi.org/10.1177/1053815117737339

Ibrahim, A. I. (2015). Effects of modelling on the teaching of pre-primary children with visual impairment. National Journal of Inclusive Education, 3(1), pp. 261-267. 

Jacobi-Vessels, J. L., Todd Brown, E., Molfese, V. J., & Do, A. (2016). Teaching preschoolers to count: Effective strategies for achieving early mathematics milestones. Early Childhood Education Journal, 44, 1-9. http://doi.org/10.1007/s10643-014-0671-4

Kapperman, G., Heinze, T., & Sticken, J. (1997). Strategies for developing mathematics skills in students who use braille. Research and Development Institute, Inc.

Karsli, E., & Allexsaht-Snider, M. (2015). Video destekli anne-baba görüşmeleri: Erken çocukluk döneminde matematik eğitimini keşfetmek için güçlü bir yöntem  [Video-cued parental dialogs: A promising venue for exploring early childhood mathematics]. Egitim Ve Bilim, 40 (179), 217-240. https://dx.doi.org/10.15390/eb.2015.4227

Klingenberg, O. G., Holkesvik, A. H., & Augestad, L. B. (2019). Research evidence for mathematics education for students with visual impairment: A systematic review. Cogent Education, 6(1). https://doi.org/10.1080/2331186X.2019.1626322

Lago, R. M., & DiPerna, J. C. (2010). Number sense in kindergarten: A factor-analytic study of the construct. School Psychology Review, 39(2), 164-180. https://doi.org/10.1080/02796015.2010.12087771

Lee, J., Kotsopoulos, D., & Stordy, C. (2012). Mathematically-relevant input during play of a caregiver with a visual impairment and her toddler. International Journal of Early Childhood, 44, 71-90. https://doi.org/10.1007/s13158-011-0040-z

Liedtke, W. (1998). Fostering the development of number sense in young children who are blind. Journal of Visual Impairment & Blindness, 92(5), 346-349. https://doi.org/10.1177/0145482X9809200513

McGuire, P., Kinzie, M. B., & Berch, D. B. (2012). Developing number sense in pre-k with five-frames. Early Childhood Education Journal, 40(4), 213-222. https://doi.org/10.1007/s10643-011-0479-4

Mon, C. S., & Yap, K. M. (2012). An investigation into virtual objects learning by using haptic interface for visually impaired children. Sunway Academic Journal, 9, 29-42. http://eprints.sunway.edu.my/390/1/SAJ_9_2012_29-42.pdf

Mononen, R., & Aunio, P. (2016). Counting skills intervention for low-performing first graders. South African Journal of Childhood Education, 6(1), 1-9. https://doi.org/10.4102/sajce.v6i1.407

Moomaw, S., Carr, V., Boat, M., & Barnett, D. (2010). Preschoolers' number sense. Teaching Children Mathematics, 16(6), 332-340.

Parks, A. N., Wakabayashi, T., & Hardin, B. (2018). Chewing on meaningful counting in prekindergarten. Teaching Children Mathematics, 25 (1), 53-59. https://doi.org/10.5951/teacchilmath.25.1.0053

Perna, R., & Loughan, A. R. (2014). The complexities of math skills development. International Journal of Mathematics, Game Theory, and Algebra, 23 (3), 165-185.

Root, J. R., Jimenez, B. A., Saunders, A. F., & Stanger, C. (2020). Replication research to support mathematical learning of students with extensive support needs. Exceptionality, 28(2), 109-120. https://doi.org/10.1080/09362835.2020.1743708

Rottmann, T., Haberzettl, N., H., & Krämer, M., (2020). Inclusive assessment of whole number knowledge—development and evaluation of an assessment interview for children with visual impairments in the primary grades. Mathematics Education Research Journal, 32(1), 147-170. https://doi.org/10.1007/s13394-019-00296-9

Segers, E., Kleemans, T., & Verhoeven, L. (2015). Role of parent literacy and numeracy expectations and activities in predicting early numeracy skills. Mathematical Thinking and Learning, 17(2-3), 219-236. https://doi.org/10.1080/10986065.2015.1016819

Smith, D. W (2017). Mathematics. In M. C. Holbrook, C. Kamei-Hannan, & T. McCarthy (Eds.), Foundations of education: Volume II: Instructional strategies for teaching children and youths with visual impairments (3rd ed.). American Foundation for the Blind Press.

Smith, D. W., & Smothers, S. M. (2012). The role and characteristics of tactile graphics in secondary mathematics and science textbooks in braille. Journal of Visual Impairment & Blindness, 106(9), 543-554. https://doi.org/10.1177/0145482X1210600905

Stipek, D. (2017). Playful math instruction in the context of standards and accountability. Young Children, 72(3), 8-12.

Trương, T. X. H. (2001). Play to form the basic math concepts in children with visual impairment. Ho Chi Minh National College of Education.

VanDerHeyden, A. M., Broussard, C., & Cooley, A. (2006). Further development of measures of early math performance for preschoolers. Journal of School Psychology, 44(6), 533-553. https://doi.org/10.1016/j.jsp.2006.07.003

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