Synaptic

Marina Paul did this work for her Honors Enrichment in my “Mathematics for Elementary Education Majors” course, as she wanted to push herself beyond the typical content of the course. Marina spent significant time researching common strategies and issues connected to teaching mathematics to students with disabilities. Her paper offers the information coming from her research, including strategies for effective teaching of mathematics to students with disabilities. Marina’s work was also informed by a couple of oral interviews she did with students with disabilities.

-Russ Goodman

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Introduction

When asked to think about learning challenges in elementary school, it would be safe to assume that many people would reflect on the difficulties of mathematics. Mathematics is a complex language of numbers, symbols, and formulae that is unlike any other subject learned in an elementary classroom, so it is understandable that many students struggle in this content area. In fact, according to Furner and Burman (2003), two-thirds of adults in the United States have a disliking and fear of mathematics (para. 4). Mathematics is especially challenging to students with disabilities, whether cognitive or physical. With the rise of research on disabilities, it is more important than ever for educators and administrators to be aware of how different disabilities affect students’s understanding of mathematics and the best teaching practices for elementary math. This project will focus on ways to improve mathematics instruction to accommodate for dyscalculia, dyslexia, mathematics anxiety, and visual impairments. Some of the information presented was informed by interviews with college students Addison VanDePol, who has been diagnosed with dyslexia, dyscalculia, and memory disorder, and Beth Rouse, who is legally blind.

Continuing with introductory comments, according to the National Center for Education Statistics (2018), “in 2015-16, the number of students ages 3-21 receiving special education services was 6.7 million, or 13 percent of all public-school students. Among students receiving special education services, 34 percent had specific learning disabilities.” In just four years, between 2012 and 2016, the number of students receiving special education services in the United States increased by 300,000 students (National Center for Education Statistics, 2018). As research and technology progress, more students are diagnosed with learning disorders, and more students with physical disabilities are able to receive special education services. These statistics can be quite overwhelming to a teacher as they are faced with an increasing number of Individualized Education Plans (IEPs) and 504 plans; differentiating instruction has become a frontrunner in lesson planning. It can be relieving for parents and students to know that teachers have a detailed plan for their child’s learning needs. However, the reality of carrying out that plan varies by teacher. One of my interviewees, Addison, remembers a frustrating discussion with a teacher regarding her 504 plan. Addison shared with me, “[the teacher] told me, ‘I don’t actually read those; I just sign them,’ so they had no idea that these are the struggles I have and here are ways to help me.” Many other students are facing this same issue of teachers not being knowledgeable on their disability and not accommodating for their learning needs. Meyen, Vergason, and Whelan (1998) state in their research that “many authorities believe that poor or traditional instruction is a primary cause of the mathematics problems of many students with learning problems” (p. 178). Teachers who lack knowledge of their students’s disabilities and ways to differentiate instruction for those students not only damage a student’s learning but also their self-worth. It is a teacher’s duty to do their best to provide students with the tools they need to succeed.

Dyscalculia

Definition and Symptoms

In their article “Treating Dyslexic and Dyscalculic Students,” Kumar and Raja (2009) define dyscalculia as “a learning disability that affects the ability to do arithmetic calculations” (p. 8). The vagueness of this definition is due to the wide variety of symptoms and difficulties within the diagnosis of dyscalculia; dyscalculia affects each person differently. Studies estimate that approximately 6% of the school-aged population has dyscalculia, which is more common than one might think when compared to dyslexia, which affects about 5% of the school-aged population (Kumar & Raja, 2009, p. 9). However, within her collection of statistics, Morin (2018) quotes a 2012 online survey done by the National Center for Learning Disabilities that found 91% of responders had heard of dyslexia and 66% had not heard of dysgraphia, dyscalculia or dyspraxia (Morin, 2018, para. 13). Some common symptoms of dyscalculia include difficulty in performing basic arithmetic skills; confusing mathematics signs and symbols; difficulty remembering mathematics concepts, rules, formulae and sequences; mistakes when reading, writing and recalling numbers, inability to understand the concepts of fractions and measurements, to organize objects in a logical way, or to comprehend financial planning or budgeting; and difficulty identifying similar signs and numerals, geometric shapes, drawing geometrical figures, conceptualizing time, or following directions (Kumar & Raja, 2009, p. 9). In their research on mathematical disabilities, Garnett and Fleischner (1987) found that early signs of dyscalculia can be found in students’s drawings of people (p. 168). Those who misrepresented the number of body parts on drawings, especially an incorrect number of fingers and an omission of the nose, were more likely to have significant mathematics achievement deficits six years later than students who do not (Garnett & Fleischner, 1987, p. 168). According to the research of Wadlington and Wadlington (2008), dyscalculia is a neurologically-based disorder, and it can be genetic (p. 2). They describe the disorder as a discrepancy between an individual’s general cognitive level and mathematical abilities (Wadlington & Wadlington, 2008, p. 2). Students with dyscalculia may be at or above grade level in other subjects; however, dyscalculia often coincides with other learning disorders.

Strategies for Instruction and Assessment

As described by King-Sears (1994) in his book on special education assessment, one of the largest debates between elementary mathematics instructors regarding students with dyscalculia is whether students with disabilities must master computational skills to progress in the mathematics curricula, or if it is a possibility to shift the focus from rote memorization to applying those computations with the use of a calculator (p. 98). As exemplified in the symptoms listed above, dyscalculia covers a broad spectrum of mathematical difficulties; however, most students with dyscalculia would benefit from the use of a calculator when attempting to apply a computation skill to a word problem because the computations are more difficult for them than the actual application (King-Sears, 1994, p. 98). King-Sears’s theory is that problem-solving could be taught preceding the introduction of a computational skill because the ultimate goal is to be able to apply those skills to real-world problems (King-Sears, 1994, p. 101). For example, one could start by prompting a student with the following problem: If Jackson has 5 apples but gives 2 apples to Noah, how many apples does Jackson have left? The student could then solve the problem by drawing a picture, using manipulatives, or visualizing taking two away from five using their fingers. Any of these strategies would provide them with the correct answer without exclusively writing out the problem in mathematical notation: 5-2=3. This strategy of teaching the problem solving first can help students understand better when they have to switch to mathematical notation. Providing them with this instruction first helps students eventually become proficient in mathematical notation. Supporters of calculator use for students with dyscalculia would agree that students need to learn and be assessed on these computational skills, but when they are being tested on the application, the use of a calculator may be in order depending on the difficulty of the computation and the needs of the specific student. In addition to calculator use, there are many other strategies that teachers should keep in mind when assessing students with dyscalculia. A current controversial topic in elementary education is the administering of timed tests. Addison mentioned in her interview that one of the accommodations listed on her 504 plan is being allowed extra time for exams (VanDePol, 2018). Having the extra time allows her to think clearly and show what she is capable of without the pressure of a strict timer. Timed tests encourage rote memorization instead of fully understanding a concept. Allowing students to have time to think about the process and demonstrate that they understand how to perform a computational skill is much more valuable than testing how fast they can read and write an answer. In addition to giving ample test taking time, measuring correct digits is a more sensitive measure of student growth than only giving credit if an entire answer is correct (King-Sears, 1994, p. 99). It is important to provide students with specific feedback as to what step they made the mistake. Another strategy for administering tests to students with dyscalculia is to break the test into two sessions or have read-aloud sections, as well as administering assessments regularly to monitor progress (King-Sears, 1994, p. 100).

In general, whether it be assessing or instructing, it is always crucial to know what is being measured. Meyen et al. (1998) give us four components of learning mathematics: problem-solving, communicating mathematics ideas, applying mathematics to everyday situations, and focusing on computational skills (Meyen et al., 1998, p. 179). When teaching students a new concept, teachers should focus their attention on what component(s) are being taught. For example, if the content involves applying mathematics to everyday situations, then it may be appropriate for a student to use a calculator since the focus is not on the computational skills. In their article focusing on suggestions for interventions for students facing mathematics difficulties, Das and Janzen (2004) provide the next key detail in the lesson planning process, which is to break the material into smaller chunks (p. 194). Because many students with dyscalculia can be weak in working memory, it is important to instruct step-by-step and take notes of the calculations of the student (Das & Janzen, 2004, p. 194). By breaking down skills into the smallest parts and presenting them step-by-step, teachers can allow students to fully process the information (Wadlington & Wadlington, 2008, p. 5). A specific strategy that teachers can demonstrate to students can be summarized by the acronym DRAW.DRAW stands for Discovering the sign; Reading the problem; Answering or drawing and checking; and Writing the answer (King-Sears, 1994, p. 102-103). Using concrete objects, pictures and diagrams can all help a student visualize the concept (Wadlington & Wadlington, 2008, p. 5).

Dyslexia

Definition and Symptoms

According to Kumar and Raja (2009), dyslexia is a language-based learning disability that includes difficulties with specific language skills, particularly reading, spelling, writing and pronouncing words (p. 8). Dyslexia also includes difficulty in comprehending text, word recognition and reading fluency (Kumar & Raja, 2009, p. 8). There is a distinction between dyslexia and dysgraphia, as dysgraphia is difficulty physically writing, often resulting in poor handwriting and some coherency issues; whereas, people with dyslexia face writing issues with spelling words or forming letters correctly. Dyslexia can be inherited or can be caused by damage to a certain part of the brain (Kumar & Raja, 2009, p. 8). Bender (2005) describes how the various regions of the brain take part in decoding language (p. 3). The visual cortex takes in information; the angular gyrus decodes sounds and processes written language; Werncike’s area comprehends language; Broca’s area finds meaning of the context of a word or number and relation to other words or numbers (Bender, 2005, p. 3-4). Though dyslexia affects reading, writing and speaking, 40% of children with dyslexia also have trouble learning mathematics (Kumar & Raja, 2009, p. 9). Garnett and Fleischner (1987) describe mathematics as “a language; a mean of representing reality through a symbol system composed of ideographs (numbers and signs) which convey concepts and ideas” (p. 168). Language processing disabilities, such as dyslexia, can make it difficult to grasp this special vocabulary of mathematics, and without an understanding of this mathematical language, it is difficult to build up a knowledge of mathematics (Kumar & Raja, 2009, p. 9).

Strategies for Instruction and Assessment

One of the largest obstacles that students with dyslexia face when doing mathematics is the written aspect, such as reading word problems and writing out long sequences of numbers. As mentioned in the dyscalculia section, it is important to keep in mind what the goal is for a lesson, or what is being measured. If the student is being assessed on mathematical content, teachers can give oral exams and read the problems aloud to the students because they are not being tested on their ability to read. Addison told the story of the first time a teacher administered her exam orally, saying, “He knew that I knew the material. I just couldn’t put it into words on paper. Saying it out loud showed him what I was really capable of” (VanDePol, 2018). Allowing students to talk out loud can also help them demonstrate whether or not they have the correct thought process for solving a mathematics problem. In addition to oral exams, teachers can also read problems aloud during instruction for students who have poor reading abilities (Wadlington & Wadlington, 2008, p. 6). Many online textbooks have read-aloud options now, so a student can listen to the problem and then solve it. This reduces the possibility of a student getting a wrong answer due to misreading the problem. Similar to testing students with dyscalculia, it may be necessary to break down a test into two sessions or to allow extra time for a student to take the exam (Wadlington & Wadlington, 2008, p. 6).

Addison also mentioned in her interview that she could not use Scantron sheets to take exams because, with her dyslexia, it was easy to accidentally skip a line or mark an answer in a wrong spot (VanDePol). Allowing a student to have extra work space is a simple thing to do that can make a big difference to a student with dyslexia or dysgraphia. Wadlington and Wadlington (2008) also suggest that students with dyslexia try writing mathematics problems on graph paper in order to align numbers more easily and to organize one’s work (p. 6).

Mathematics Anxiety

Definition and Symptoms

According to Furner and Burman (2003), mathematics anxiety is defined as an “irrational dread of mathematics that interferes with manipulating numbers and solving mathematical problems within a variety of everyday life and academic situations” (para. 2). It is comparable to test anxiety as it involves cognitive concerns about performance and heightened emotional reactions to stress; however, it is not strictly confined to testing and only affects the learning of mathematics, not other subjects (Furner & Burman, 2003, para. 9). Mathematics anxiety is not officially categorized as a learning disability, but it greatly interferes with the ability to learn mathematics and perform on tests (Wadlington & Wadlington, 2008, p. 3).

Mathematics anxiety can be caused by a variety of factors including parents who have negative attitudes toward mathematics, teachers who are not confident in mathematics, poor instruction, as well as strategies such as assigning the same work for all students, covering a textbook problem by problem, giving written work every day, teaching only one way to complete a problem, and assigning mathematics problems as a punishment for misbehavior (Furner & Burman, 2003, para. 7). Many students struggle with a fixed mindset about who is good at mathematics and who is not. If students perceive themselves as not good in mathematics, they can develop a fearful avoidance of mathematics (Wadlington & Wadlington, 2008, p. 3). Up to around age 12, male students tend to have more mathematics anxiety, but as females enter the late junior high and early high school phase, many begin to have higher levels of mathematics anxiety than males (Furner & Burman, 2003, para. 7). Mathematics anxiety heavily impacts a students’s perception of their mathematical abilities, and if it develops in a student, it is crucial that teachers are aware of the best teaching practices to help the student find mathematical success.

Strategies for Instruction and Assessment

Many students develop mathematics anxiety from watching parents or teachers who are not confident in mathematics or who dislike of the subject. It is crucial that in both mathematics instruction and assessment, teachers are confident and positive. When initially assessing a student with mathematics anxiety, teachers should be aware of the following dispositions: confidence in using math; flexibility in exploring mathematical ideas; willingness to persevere in mathematical tasks; interests, curiosity and inventiveness in doing math; ability to reflect on and monitor their own thinking when it comes to math; and focus on value of and appreciation for mathematics (Furner & Burman, 2003, para. 2-3). In order to improve students in these areas of mathematical learning, teachers must demonstrate an appreciation of math. It is always important to explain to students the relevance of the mathematics content and how it applies to real life. Students also need to see that it is not only acceptable but encouraged to make and learn from mistakes in order to develop a growth mindset, in which they believe that their learning can expand from where they are at currently. This is all part of creating a safe classroom environment for learning mathematics (Wadlington & Wadlington, 2008, p. 5). Teachers should never punish a student for an incorrect answer, and students should never receive ridicule from either the teacher or other students for an incorrect answer (Wadlington & Wadlington, 2008, p. 4). A classroom needs to be perceived by students as a safe environment in which to grow. Addison discussed in her interview the ugly reality of her elementary schooling experience; she felt as though many of her teachers did not care for her success (VanDePol, 2018). Her teachers never tracked her progress from the standpoint of a student with a disability, and she could not remember any time when a teacher accommodated for her learning needs during elementary school (VanDePol, 2018). It makes the world of a difference when a teacher acknowledges that a student is struggling and advocates for accommodations for that student. It is also important that students hear praise frequently for both small and large successes (Wadlington & Wadlington, 2008, p. 4).

Teachers need to recognize students who may have mathematics anxiety and find ways to make mathematics lessons engaging for them. Sharing personal stories related to mathematics content and playing mathematical games are both ways to help foster good attitudes about mathematics (Wadlington & Wadlington, 2008, p. 4). Teachers can also make lessons engaging by emphasizing class discussion so that students can hear and learn from their peers as well (Furner & Burman, 2003, para. 16). Another great idea that Furner and Burman (2003) share is hosting family mathematics nights at school. Having parental involvement can change a student’s attitude toward mathematics, and “mathematics night” can help parents learn about the content their students are learning and how teachers are teaching it (Furner & Burman, 2003, para. 28-29).

Visual Impairments (Blindness)

Definition, Types of Blindness and Effects on Mathematics Education

When most people hear “blindness,” they typically think of complete blindness, or a complete loss of vision. However, according to the Healthery website (2017), “blindness simply means any condition where insufficient light is reaching the retina, or the electrical synapses either aren’t firing correctly, or the information isn’t being delivered correctly to the brain” (para. 2). There are multiple types of visual impairment, including legal blindness (complete and near-complete blindness), achromatopsia (or color blindness), and night blindness. Being considered legally blind requires a person to have vision equal to or less than 20/200 in his or her better eye, meaning they can see object no greater than 20 feet away.

Being completely blind means have a complete loss of vision; however, “many people who are legally blind can still see light, shapes and even some objects from a close distance” (Healthery, 2017, para. 5). Some cases of complete or near-complete blindness are caused by a congenital birth defect (Healthery, 2017, para. 6). I had the opportunity to interview Beth, who is legally blind but not completely blind. Beth explained to me that her “eyesight is a genetic disease, which basically means that [she] inherited a recessive gene from both of [her] parents” (Rouse, 2018). Beth is not completely blind, meaning she can still see some shapes when close to her eyes. For students like Beth, their blindness is often diagnosed as infants. Babies typically track their parents with their eyes based on what they hear and what they see. Beth was diagnosed at five months when a daycare provider noticed that she was not tracking people with her eyes if they did not say anything. Other causes of complete or near-complete blindness include trauma, injury, progressive disorders, cataracts, and one of the leading causes is diabetes (Healthery, 2017, para. 6,7). Color blindness, or achromatopsia, “relates to a person not being able to distinguish between colors effectively” (Healthery, 2017, para. 8). Rarely are people fully color blind; they more often have trouble distinguishing between certain colors (Healthery, 2017, para. 8). For example, one of the most common types of color blindness is red-green blindness, meaning these people have trouble distinguishing between red and green and shades of both colors.

Color blindness is caused by the lack of a third cone within the eye, either red, green or blue, or the malfunction of the cones (Healthery, 2017, para. 9). Color blindness is a genetic disorder, and occurs more commonly in males than females because it is carried within the X chromosome. Because females would need both X chromosomes to carry the gene, it occurs less likely than in males who only need the one X chromosome to carry the gene (Healthery, 2017, para. 10). Students with color blindness may experience minor difficulties distinguishing between figures or graphs if color is the characteristic factor.

Night blindness refers to a “visual impairment present in dim light (at night-time or in a dark room)” (Healthery, 2017, para. 12). This can be caused by a defect in the natural way the eye adjusts to levels of brightness, a Vitamin A deficiency, birth defect, trauma, glaucoma, etc. (Healthery, 2017, para. 13). Night blindness would most likely not affect a students’ learning of mathematics.

Because learning mathematics is a highly visual process, it is clearly a difficult subject to teach students with visual impairments. Herzberg and Rosenblum (2011) say that “students with visual impairments (that is, those who are blind or have low vision) face challenges as they work toward mathematical competence, because the eye can more quickly and efficiently take in pieces of information simultaneously than can the fingers” (Herzberg & Rosenblum, 2011, para. 3). Unlike the other disabilities described thus far, visual impairment does not affect a student’s ability to understand a concept, but differentiated instruction becomes increasingly important. Beth shared that the following quote:

I just got used to the fact that math took me two to three times longer than everyone else, and it was never a question of ‘do you understand what’s going on?’ I clicked onto the concepts just as fast as other people. It’s just a matter of reading what was on the board or getting my thoughts out of my head and onto paper.

Strategies for Instruction

Hearing Beth’s story, many teachers would not know how to best teach her. Instructing students affected by visual impairments requires a unique approach. The following strategies are just a few of the many ways to best teach students with visual impairments.

Springboard (1997) discusses the importance of advocating for students with a visual impairment. Instead of a tutor, it is more beneficial for a student to have a facilitator, who helps support the instruction of the classroom teacher by adapting the way the information is presented during class or soon after (p. 79). This facilitator’s job includes consulting with the teacher to create a more inclusive classroom environment for the student with a visual impairment, serving as a liaison between the school and parents, and promoting the acceptance of and providing advocacy for the students (Springboard, 1997, p. 79). Beth recalled memories with her araeducator Bonnie, whom she had from kindergarten through her senior year of high school. Students with visual impairments will also often receive services from an orientation mobility specialist or a teacher for the visually impaired (TVI) to learn skills, such as using a cane or practicing Braille, to help them become more independent people and learners (Rouse, 2018). It is imperative that general education teachers collaborate with all of these specialists, especially the paraeducator who can help during mathematics instruction. For example, Beth told me about how Bonnie would help explain what was written on the board as well as translate worksheets into Braille (Rouse, 2018).

Because mathematical images can be complex, especially for students with visual impairments, it is necessary to break down diagrams into parts so that students can form a mental image (Herzberg and Rosenblum, 2011, para. 4). For students like Beth who have some vision, having a small white board specifically for her can be incredibly helpful so that she can hold it up close to visualize shapes (Rouse, 2018). Students who have visual impairments are tactile learners, meaning they learn using what they can feel versus what they can see. Beth shared with me that learning mathematics was made possible for her by making content as three-dimensional as possible (Rouse, 2018). Herzberg and Rosenblum say that “accurate and well-prepared tactile materials are essential in a student’s success in math” (2011, para. 4). It is necessary for teachers of students with visual impairments to become proficient in tactile instruction; they can attend training sessions through university teacher prep program courses, workshops and conference sessions (Herzberg & Rosenblum, 2011, para. 5). Using a Braille reader, students can decipher mathematics problems using what is called Nemeth code (Springboard, 1997, p. 78). Using “finger math” or other tactile manipulatives in instruction can also increase computation accuracy (Ferrell, 2006, para. 13). Other important accommodations for students with visual impairments include giving extra time on assessments, distributing textbooks ahead of time to be translated into Braille, and providing ample materials for successful learning in the general education classroom (Rouse, 2018).

Teaching mathematics to students with visual impairments often requires the use of technology that allows the students to read and solve mathematics problems. With many of the cognitive learning disabilities above, using pictures and visuals can aid in instruction, but for students with visual impairments, instruction relies on how well content is explained either orally or through writing. One example of technology that made an impact on the education of students with visual impairments is the APH peg slate, which allows students to read, solve, and record answers to mathematics problems in Braille (Springboard, 1997, p. 78). Beth informed me of three forms of technology she uses (Rouse, 2018). One is her Braille Note, which is essentially a computer using Braille because it has access to the internet, and it has a calculator. Another is a Perkins Brailler, which she compared to a typewriter, as it physically punches the Braille into paper, rather than an electronic copy like the Braille Note. The last is a peg slate, which is a non-electronic device she uses occasionally to write in a Braille on paper. The peg slate requires one to write in Braille backwards. Overall, successfully teaching students with visual impairments is made possible by communicating with the students, parents, and specialists.

Conclusion

According to the National Council of Teachers of Mathematics, “all students can learn mathematics if given high-quality instruction, challenging content, and appropriate accommodations, and well-prepared, knowledgeable teachers must have adequate resources and support to deliver high-quality instruction” (Herzberg & Rosenblum, 2011, para. 2). If given the best instruction that fits their needs, every child can succeed in mathematics. When teachers are confident in mathematics, use visual learning tools, break down concepts, provide accommodations and implement effective mathematics interventions, students with disabilities, such as dyscalculia, dyslexia, mathematics anxiety, and visual impairments, can achieve mathematical success in the general education classroom. The mean mathematics scores of students with learning disabilities in the 12th grade is a high 5th grade level (Meyen et al., 1998, p. 178). Teachers have the ability and responsibility to change this statistic by making changes in their instruction style and by seeking out the resources necessary to help every child succeed. Mathematics is a difficult subject, especially for students with disabilities, but well-prepared and knowledgeable teachers can make learning mathematics possible.

Works Cited

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VanDePol, A. (2018, October 17). Personal interview.