# Effective Methods for Delivering Mathematics Instruction to Students with Visual Impairments

**By Anne Brawand and Nicole Johnson**

*Dr. Anne Brawand, Department of Special Education, Kutztown University of Pennsylvania*

*Dr. Nicole Johnson, Department of Special Education, Program of Visual Impairments, Kutztown University of Pennsylvania*

## Abstract

Schools throughout America put an emphasis on instruction in mathematics. Students who are visually impaired should not be left out of this national effort (Kapperman & Sticken, 2003). It has been established that children who are visually impaired should learn mathematical skills at the same level as their sighted peers (Tindell, 2006). However, the acquisition of mathematical skills can be more difficult for students with visual impairments due to the abstract nature of many essential concepts and the highly visual presentation of the subject (Kapperman, Heinze, & Sticken, 2000). This discussion highlights the most effective methods for delivering mathematics instruction to students with visual impairments, based on a summary of literature. Implications for teacher implementation in the field are also provided.

**Keywords **

Visual impairments, mathematics skills, methods

## Introduction

The No Child Left Behind Act of 2001 (NCLB) established accountability measures including annual assessment of students in the area of mathematics. Proficiency in mathematics is necessary because it is critical to functioning adequately in the context of daily life situations (Jitendra, George, Sheetal, & Price, 2009). The National Mathematics Advisory Panel (2008) reported that mathematical problem solving is one of the most important concepts taught in the grade-level curriculum. Legislation and state testing requirements such as NCLB’s state assessments (2001) also focus on a conceptual understanding of mathematical concepts, skills, and relations of numbers (Jitendra, 2007).

Thorough understanding of mathematics enhances educational and occupational opportunities for all people. The workplace also requires increasingly advanced computational and technological skills; those who do not possess these skills are restricted in career choices (Kapperman & Sticken, 2003). Vision allows access to information that helps build a conceptual understanding of basic mathematical concepts. Consequently, students with visual impairments tend to face more challenges than their peers without disabilities, even when learning the most basic mathematical concepts (Beal & Shaw, 2008). These difficulties include problem solving, gaining access to the problem information, representing problem information, and calculating the answer. In addition, mathematics is highly visual in nature and often uses graphics to convey important information, presenting an additional obstacle for students with visual impairments (Smith & Smothers, 2012).

Some challenges that students with visual impairments encounter when learning mathematics can be overcome when the content is taught in an appropriate manner, such as by using programmed instruction (Agrawal, 2004). Programmed instruction, which includes developing a sequence of instructional activities, has the potential to maximize learning and increase motivation for students with visual impairments. It has also been established that children who are visually impaired should learn mathematical skills at the same level as their sighted peers (Tindell, 2006). An opportunity to explore mathematics using different instructional tools in these early years is beneficial to this population of students as well (Amato, Hong, & Rosenblum, 2013). Finally, exposing students with visual impairments to experiences in individual environments promotes natural development of mathematics skills, which can lead to positive attitudes toward mathematics and build students’ confidence level (Research and Development Institute, Inc., 2006).

The following discussion highlights the most effective methods for delivering mathematics instruction to students with visual impairments, based on a summary of the literature. These methods include using a combination of the abacus, braille codes, tactile materials, and concrete materials to teach mathematics skills to students with visual impairments. The use of the braille Nemeth code and abacus has long been established as a beneficial practice to teach mathematical concepts to students with visual impairments (Kapperman et al., 2000). The braille Nemeth code gives students who are blind the opportunity to read and write braille mathematical computations, helping to build a thorough understanding of concepts presented. The abacus is a useful calculation tool whether used alone or in conjunction with other devices because of its speed, accuracy, portability, and flexibility (Kapperman, Heinze, & Sticken, 1997). Because students with visual impairments may take longer to learn mathematical concepts, it is essential for teachers of students with visual impairments (TVIs) to learn how to utilize a variety of tools to complete computations. Therefore, university preparation will also be discussed, followed by implications which teachers should consider when implementing these strategies in the field.

## Use of the Abacus

The Cranmer abacus is one tool used to teach students with visual impairments to compute mathematical problems. According to Amato et al. (2013), the operations of addition and subtraction are the most frequently taught skills using the abacus. Setting and counting are also in the realm of mathematical skills taught using an abacus. Conversely, higher-level mathematical computation skills, such as multiplication with multiple digits and computing with fractions, were reported to be taught less frequently using an abacus. In addition, TVIs do not use the abacus for these higher order skills because of the increased use of technology to solve problems in higher grades, students’ use of mental math, and the abacus not being deemed an appropriate tool for certain students. The counting method was found to be the most widely used for abacus instruction. In the counting method, the student counts each bead as it is added or subtracted, moving from the unit beads to the five beads with 1:1 correspondence. The logic-partner method is another method found to be used frequently as it focuses on understanding the “what” and “why” of the steps in solving a problem on the abacus using synthesis and verbalizing. Finally, most TVIs begin instruction with the abacus when their students are between preschool and second grade.

Amato et al. (2013) report that students who have abacus skills included in their Individualized Education Program (IEP) seem to enjoy instruction and experience positive results when the classroom teacher is able to learn with the student. However, many students with visual impairments do not have IEP goals related to use of the abacus. Even when listed as an accommodation, students with visual impairments tend not to use the abacus after fifth grade, and prefer to use the talking calculator in high school (Amato et al., 2013). On the other hand, students must know how to add, subtract, multiply, and divide in order to use the abacus; therefore, it is not equivalent to a calculator. Surprisingly, there are some TVIs that do not implement abacus instruction for different reasons. Amato et al. (2013) reported that some TVIs do not teach abacus skills to children with visual impairments because they: (a) do not believe they are effective instructors of abacus; (b) lack confidence; (c) do not know where to get updated training; (d) do not have sufficient time to teach students due to curricular demands; and (e) increase the use of technology, which is the students’ preferred method of learning. However, it is important to remember that a teacher’s competence and attitude have a great impact on a student’s potential for success with the Cranmer abacus (Kapperman et al., 2000, p. 385).

## Braille Use

The Nemeth Code for braille mathematics and science notation is also an effective tool which gives students who are visually impaired access to mathematical work, and allows them to produce solutions to computations (Rosenblum & Smith, 2012). Dr. Abraham Nemeth developed the Nemeth Code to introduce principles and procedures for the presentation of braille equivalents for the complex signs and configurations of ink-print mathematical and scientific notation (Craig, 1987). This code is a living document and continues to be updated and refined to assure the correct transfer from ink print to braille.

As the United States transitions to Unified English Braille (UEB), TVIs need to be proficient in both codes in order to be able to convert efficiently from one code to the other. Without knowledge of these codes, students who are visually impaired are not able to participate with their peers in mathematics. When teaching mathematical concepts, TVIs should ensure the student is presented with flawless braille and adhere to all of the mathematical code’s rules (Kapperman et al., 2000). Considerable confusion can arise when changes to the written forms of the symbols are made, which means that the correct form should always be maintained. When providing instruction in advanced mathematics, TVIs are not expected to know the meaning of each print symbol, but should consult with the general education teacher to ensure that the symbols have been correctly interpreted by the student who is blind. Overall, TVIs need to feel confident in their ability to teach the Nemeth Code and UEB to verify that students are accessing all areas of mathematics and science properly.

## Tactile Graphics

Tactile graphics, or “graphics intended to be read principally by touch rather than vision,” are also a beneficial tool for teaching mathematics to students with visual impairments (Aldrich, Sheppard, & Hindle, 2003, p. 284). Tactile graphics represent a variety of print illustrations that contain information given in graphic formats. They are just as important to the braille reader as print diagrams are to the sighted student (O’Day, 2014). These representations of print graphics are designed in a way that is most meaningful to the reader (Braille Authority of North America, 2012) and are produced using a variety of materials and methods. Nonetheless, a major issue with tactile graphics that students with visual impairments must overcome is the amount of visual aspects in mathematics, specifically in geometry and data analysis (Smith & Smothers, 2012). Due to the inclusion of these highly visual concepts, it is important that tactile graphics are utilized to develop a thorough understanding in mathematics for students with visual impairments.

In addition to providing tactile graphics for their students, it is necessary that TVIs teach students how to read and make sense of tactile graphics in mathematics instruction. In order to teach students with visual impairments to be successful in handling and interpreting a variety of tactile graphics, TVIs should use a sequence for introduction of tactile graphics (Koenig & Holbrook, 2000). TVIs could first present students with opportunities to handle real objects, transition to the use of models, and finally implement two dimensional representations. The successful reading of tactile graphics requires knowledge of spatial and geographic concepts and strategies for exploring and interpreting the displays (Kapperman et al., 2000). Through a survey of TVIs, Zebehzazy and Wilton (2014) found that the majority of respondents acknowledged their roles in teaching students how to use graphics and felt it was necessary to teach students specific strategies for working with graphics in order for them to gain the most information. The use of tactile graphics in mathematics for students with visual impairments can assist the students in building a greater understanding of visual mathematical concepts that otherwise would be misunderstood.

## Concrete Materials

Concrete materials, or manipulatives, are available in multiple forms. They are often defined as “physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics (TeacherVision, n.d.).” Manipulatives assist students in learning numerous mathematical concepts including: addition and subtraction, operations with integers, fraction equivalents, counting money, telling time, measurement, and place value (Mastropieri & Scruggs, 2010). Manipulatives also offer children with visual impairments concrete experiences to help them understand their environment and learn concepts parallel to sighted peers in the classroom setting (Saracho, 2012). Concrete manipulatives are one of the major tools used when teaching visually impaired students during initial instruction in mathematical calculations as well (Kapperman et al., 2000). Finally, concrete manipulatives increase computation accuracy and have been recommended when teaching students with visual impairments for at least 30 years (Belcastro, 1993; Champion, 1977; Hatlen, 1996).

A variety of sophisticated aids, teacher-made aids, and real objects/materials in the environment can be implemented as concrete manipulatives when teaching students with visual impairments (Mani, Plerchaivanich, Ramesh, & Campbell, 2005). Real objects can be used to sort, match, compare, group, take apart, put together, seriate, and count. When teaching various mathematical concepts, items that can be used to help students with visual impairments conceptualize what is being taught include: various geometric shapes of different sizes, textures, and colors; felt boards; symbol stickers; and base ten blocks. Through the use of concrete manipulatives, students can more easily remember what they did and explain concepts to solve the problem (Stein & Bovalino, 2001).** **Manipulatives use also benefits students without visual impairments in the area of mathematics (Miller & Hudson, 2007). The National Council of Teachers for Mathematics recommends the use of manipulatives because it is supported by both learning theory and educational research in the classroom. The effective use of manipulatives assists students with connecting ideas and integrating their knowledge so they gain a deeper understanding of mathematical concepts (Research on the Benefits of Manipulatives, 2009).

To develop conceptual understanding using concrete manipulatives, students with visual impairments should read a mathematical problem, write it, listen to it being read, tactually explore the problem through manipulatives, and if possible move their body and/or manipulative through space (Osterhaus, 1996). It is essential to use concrete materials appropriately when teaching new concepts by allowing students to explore the manipulative initially, and then integrate it into the curriculum. Choosing the correct manipulative for instruction is also an important task. Manipulatives should be aligned to the specific goals and skills of the mathematical program (Boggan, Harper, & Whitmire, 2010; Brawand, Johnson, & Kolvites, 2015). The complexity of the manipulatives provided should increase as the students’ thinking and understanding of mathematical concepts increase (Seefeldt & Wasik, 2006). An intervention that integrates the use of manipulatives into explicit instruction designed to teach important concepts is the Concrete-Representational-Abstract (CRA) method (Miller & Hudson, 2007). CRA has been effective in improving mathematical skills for students with and without disabilities in the general education classroom. When the TVI and general education teacher use creativity in the selection of materials and interventions, they can bring utmost instructional variety to the child with visual impairments (Mani et al.,** **2005).

## University Preparation

Throughout schools in America, an emphasis is placed upon mathematics instruction. Students with visual impairments should not be disregarded in this effort. Kapperman and Sticken (2003) recommend that personnel preparation programs for training TVIs place a greater emphasis on training in braille mathematics to ensure that students with visual impairments are not left out. TVIs are responsible for teaching students specialized computation methods including the use of the Nemeth Code, an abacus, and a braillewriter. It is expected that TVIs provide appropriate mathematics instruction and monitor that students with visual impairments make steady gains in acquiring mathematical concepts. Additionally, TVIs and general education teachers are expected to work as a team in order for students with visual impairments to meet educational milestones in mathematics (Research and Development Institute, Inc., 2006). The TVI brings knowledge about adaptations and accommodations for students with visual impairments while the general educator is the specialist in mathematics content. In addition to collaborating with general educators, TVIs should use formal and informal assessment strategies, teach specialized computation methods and braille mathematical codes, demonstrate how to interpret and utilize tactile graphs, and provide modified learning materials for mathematics instruction (Kapperman et al., 2000, p. 372).

Rosenblum and Smith (2012) conducted a study on mathematics instruction using specialized braille codes, abacus, and tactile graphics at universities in the United States and Canada. They found that all 26 university programs sampled provided instruction in the Nemeth Code. This study also found that university programs required students to prepare tactile graphics, and 25 out of 26 programs taught students how to compute with an abacus. However, less than half of the universities expected students to master advanced mathematical symbols, such as lines and arcs. It is essential that university-based preparatory TVI programs place an emphasis on teaching braille mathematics and the application of all the tools available to meet students’ needs in relation to mathematics. Spungin and Ferrel (2007) described the role of TVIs as multifaceted. Consequently, TVIs must be well prepared in all areas to meet the diverse needs of their students and feel confident in doing so.

## Implications for Practice

Students with visual impairments require a thorough understanding of mathematics to function in today’s society. It is beneficial for mathematics to be presented to students with visual impairments using a combination of the abacus, braille codes, tactile materials, and concrete materials in order to meet academic goals. Teaching of mathematical concepts through the use of abacus, braille codes, manipulatives, tactile graphics, and hands-on experiences should start in the early years (Amato et. al, 2013). As soon as a child begins to learn braille, practice in braille mathematical codes should be given. Experiences have demonstrated that children with visual impairments display resentment to learning mathematical codes when introduced at later stages (Mani et al.,** **2005). Whenever a sighted student is allowed to use scratch paper for solving problems, the use of the abacus should be permitted for students who are visually impaired (Amato et al., 2013).The abacus can be taught in conjunction with braille mathematical codes to help students conceptualize concepts in mathematics. It is important for TVIs to be highly proficient in the use of abacus computation skills, braille mathematical codes, and tactile graphics and adapting the general education mathematics curriculum to meet individual student needs.

Due to the limited time in personnel preparation programs, universities should assure that future TVIs have at least an awareness of the variety of methods and resources which are available for self-study (Rosenblum, Hong & Amato, 2013). When students are braille readers, it is imperative that TVIs are proficient in the use of the Nemeth Code for mathematics and science notation in order to teach their students to read and write the symbols that underpin the study of mathematics (Kapperman, 1994). The general education teacher, TVI, paraprofessional, family, and visually impaired student also need to collaborate in order for the student to fully benefit from mathematics instruction in the classroom (Research and Development Institute, Inc., 2006). In light of this situation, students who are visually impaired need not be left out of the national effort to emphasize mathematics instruction in schools throughout America (Kapperman & Sticken, 2003).

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