Arithmetic involves recognizing numbers and symbols, memorizing facts, aligning numbers, and understanding abstract concepts such as place value and fractions. Any of these may be difficult for children with developmental arithmetic disorders, also called dyscalculia, which refers to selective impairment in mathematical thinking or in calculation skills (Fletcher & Forman, 1994). Problems with number or basic concepts are likely to show up early. Disabilities that appear in the later grades are more often tied to problems in reasoning.
Various types of mathematical disorders have been cited by the American Academy of Special Education Professionals' Educator's Diagnostic Manual of Disabilities and Disorders (in press). Below are mathematical disorders frequently seen in children with dyscalculia:
Basic Number Fact Disorder
Individuals with a Basic Number Fact Disorder have problems memorizing and retaining basic arithmetic facts, such as the answers to 8 – 2, 7 + 1, or 12 ˘ 2. It is not that individuals with Basic Number Fact Disorder do not remember any arithmetic facts, but rather they have problems memorizing as many facts as other children do. Furthermore, they appear to forget facts rather easily. These children may struggle for years, will count their fingers to add and subtract, and seem unable to develop efficient memory strategies on their own.
By definition, calculation is problem solving that involves numbers or quantities. The calculation of numbers often gives students with learning disabilities great difficulties. Inconsistent calculation can lead to numerous errors when doing math work. Students with calculation difficulties often perform the incorrect mathematical operations. For example, when calculating 8 + 2, they may respond , they subtracted rather than added the two numbers.
Mathematical Abstraction Limitation Disorder
Individuals with this disorder do not possess the ability function at a high level of mathematical abstraction and as a result can only function on a concrete level of understanding. Individuals with this disorder tend to reach a ceiling in their ability to comprehend abstract math concepts.
Mathematical Estimation Disorder
Children with dyscalculia seem to have an impaired sense of number size. This may affect tasks involving estimating numbers in a collection and comparing numbers.
Mathematical Language Disorder
According to Garnett (1998), some students with LD are particularly hampered by the language aspects of math, resulting in confusion about terminology, difficulty following verbal explanations, and weak verbal skills for monitoring the steps of complex calculations. Teachers can help by slowing the pace of their delivery, maintaining normal timing of phrases, and giving information in discrete segments. Such slowed-down chunking of verbal information is important when asking questions, giving directions, presenting concepts, and offering explanations.
Mathematical Measurement Disorder
Individuals with this disorder may have difficulty with concepts involving measurements, such as speed (miles per hour), temperature (energy per unit of mass), averages, and proportional measures.
Mathematical Navigation Disorder
Children with this disorder can usually learn the sequence of counting words, but may have difficulty navigating back and forth, especially in twos, threes, or more.
Mathematical Organization Disorder
Individuals with this disorder may have an inability to organize objects in a logical way. They may be unable to comprehend or mentally picture mechanical processes. They may lack "big picture/whole picture" thinking. They may have a poor ability to visualize the location of the numbers on the face of a clock, the geographical locations of states, countries, oceans, streets, and so on.
Mathematical Sequencing Disorder
People with this disorder have trouble with sequence, including left/right orientation. They will read numbers out of sequence and sometimes do operations backwards. They also become confused on the sequence of past or future events.
Symbolic Mathematical Operations Disorder
Individuals with this disorder may find it especially difficult to translate between number words, where powers of ten are expressed by new names (ten, hundred, and thousand) and numerals (where powers of ten are expressed by the same numerals but in terms of place value).
Temporal/Monetary Math Disorder
People with this disorder tend to have difficulties in topics relating to time, telling time, keeping track of time, estimating time, monetary concepts, and counting money. Older children may exhibit difficulties with money and credit and cannot do financial planning or budgeting (e.g., balancing a checkbook). Individuals may have fear of money and cash transactions and may be unable to mentally figure change due back, the amounts to pay for tips, taxes, and so forth.
Visual–Spatial Math Disorder
Students with this disorder have disturbances in visual–spatial–motor organization, which may result in weak or missing understanding of concepts, very poor number sense, specific difficulty with pictorial representations, poorly controlled handwriting, and confused arrangements of numerals and signs on the page. Students with this disorder might have spatial problems and difficulty aligning numbers into proper columns.
Written Symbol System Disorder
According to Garnett (2000), many younger children who have difficulty with elementary math actually bring to school a strong foundation of informal math understanding. They encounter trouble in connecting this knowledge base to the more formal procedures, language, and symbolic notation system of school math (Allardice & Ginsburg, 1983). The collision of their informal skills with school math is like a tuneful, rhythmic child experiencing written music as something different from what she already can do. In fact, it is quite a complex feat to map the new world of written math symbols onto the known world of quantities, actions and, at the same time, to learn the peculiar language we use to talk about arithmetic.
Whether because of the reading requirement or the ability to understand the mathematical concepts captured in a problem, students with learning disabilities may be unable to sort critical extraneous information, to recognize the correct computational procedure, or to determine whether the answer they obtain is reasonable (Jordan & Hanich, 2003). Mathematical difficulties are often major obstacles in the academic paths of students with LD and frequently continue to cause problems throughout high school. Mastery of fundamental quantitative concepts is vital to learning more abstract and complex mathematics, a requirement for youth with learning disabilities who are seeking to complete high school and attend colleges or universities (Cirino, Morris, & Morris, 2002; cited in Hardman et al., 2005). Further research on difficulties with mathematics and on effective instruction for students encountering such problems grows more important as such young people seek to achieve more challenging educational goals.
Given these difficulties, it is not surprising that 50% of students with learning disabilities have IEP goals in math. As with reading and writing, explicit, systematic instruction that provides guided meaningful practice with feedback usually improves the math performance of students with learning disabilities (Fuchs & Fuchs, 2001; cited in Heward, 2003).
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