What is Mathematics?
Major Unifying Themes in This Document
Syllabus
Foundational Information
Learning Theories
Mind and Body Tools
Science of Teaching & Learning
ProjectBased Learning
Computational Mathematics
The Future
Recommendations
References
Website Author
"Dr. Dave" Moursund

References
Garnett, Kate (November, 1998). Math Learning Disabilities. Division for Learning Disabilties Journal of CEC. Accessed 9/7/03: http://www.ldonline.org/ld_indepth/math_skills/garnett.html
While children with disorders in mathematics are specifically included under the definition of Learning Disabilities (Federal Register, August 23, 1977), seldom do math learning difficulties cause children to be referred for evaluation. In many school systems, special education services are provided almost exclusively on the basis of children's reading disabilities (Badian, 1983). Even after being identified as learning disabled (LD), few children are provided substantive assessment and remediation of their arithmetic difficulties (Goodstein & Kahn, 1974).
This relative neglect might lead parents and teachers to believe that arithmetic learning problems are not very common, or perhaps not very serious. However, approximately 6% of schoolage children have significant math deficits (Kosc, 1974; Badian, 1983) and among students classified as learning disabled, arithmetic difficulties are as pervasive as reading problems (Badian, 1983; McKinney & Feagans, 1980). This does not mean that all reading disabilities are accompanied by arithmetic learning problems, but it does mean that math deficits are widespread and in need of equivalent attention and concern.
This is a moderately recent article, but it feels to me to be woefully out of date relative to current literature on how children learn to read and on reading disabilities. It does not contain information about brain mechanisms involved in learning and doing math.
Geary, David (1999). Mathematical Disabilities: What We Know and Don't Know. Accessed 9/7/03: http://www.ldonline.org/ld_indepth/math_skills/geary_math_dis.html
Over the past several decades important advances have been made in the understanding of the genetic, neural, and cognitive deficits that underlie reading disability (RD), and in the ability to identify and remediate this form of learning disability (LD). Research on learning disabilities in mathematics (MD) has also progressed over the past ten years, but more slowly than the study of RD. One of the difficulties in studying children with MD is the complexity of the field of mathematics. In theory, MD could result from difficulties in the skills that comprise one or many of the domains of mathematics, such as arithmetic, algebra, or geometry. Moreover, each of these domains is very complex, in that each has many subdomains and a learning disability can result from difficulties in understanding or learning basic skills in one or several of these subdomains.
As an example, to master arithmetic, children must understand numbers (e.g., the quantity that each number represents), counting (there are many basic principles of counting that children must come to understand), and the conceptual (e.g., understanding the Base10 number system) and procedural (e.g., borrowing from one column to the next, as in 439) features involved in solving simple and complex arithmetic problems. A learning disability in math can result from difficulties in learning any one, or any combination, of these more basic skills. To complicate matters further, it is possible, and in fact it appears to be the case, that different children with MD have different patterns of strengths and weakness when it comes to understanding and learning these basic skills.
…
It appears that many  perhaps more than 1/2  children with MD also have difficulties learning how to read and that many children with RD also have difficulties learning basic arithmetic. In particular, children and adults with RD often have difficulties retrieving basic arithmetic facts from longterm memory. The issue is whether the cooccurrence of RD and difficulties in remembering arithmetic facts are due to a common underlying memory problem. The answer to this question is by no means resolved. Nonetheless, some evidence suggests that the same basic memory deficit that results in common features of RD, such as difficulties making lettersound correspondences and retrieving words from memory, is also responsible for the factretrieval problems of many children with MD. If future research confirms this relationship, then a core memory problem that is independent of IQ, motivation and other factors, may underlie RD and at least one form of MD.
Mathematics and Dyslexia. Perspectives, Fall 1998. Accessed 9/7/03: http://www.ldonline.org/ld_indepth/math_skills/ida_math_fall98.htm
Not all individuals with dyslexia have problems with mathematics, but many do. There are those who have a good memory for sequences and can execute procedures in a "recipe" style, i.e., stepbystep. They are able to remember formulas, but may not understand why the formula makes sense. They prefer to do paper and pencil tasks and are attentive to the details, but do not see the big picture. Then, there are those who see the big picture and have insight into the patterns of mathematics, but are poor at computation and have problems with remembering stepbystep procedures. They also understand mathematical concepts and like to solve problems mentally and quickly, yet their answers may be inaccurate. These individuals may have difficulty in verbalizing and explaining their answers.
Too frequently and too readily, individuals with dyslexia who have difficulty with mathematics are misdiagnosed as having dyscalculia  literally trouble with calculating, a neurologically based disability. True dyscalculia is rare (Steeves, 1983).1 We know that for individuals with dyslexia, learning mathematical concepts and vocabulary and the ability to use mathematical symbols can be impeded by problems similar to those that interfered with their acquisition of the written language (Ansara, 1973).2 Additionally, we know that the learning of mathematical concepts, more than any other content area, is tied closely to the teacher's or academic therapist's knowledge of mathematics and to the manner in which these concepts are taught (Lyon, 1996).3 Therefore, there are individuals with dyslexia who will exhibit problems in mathematics, not because of their dyslexia or dyscalculia, but because their instructors are inadequately prepared in mathematical principles and/or in how to teach them.
…
To assist individuals with dyslexia in making this linkage, it is essential that teachers and academic therapists provide instruction that allows the learner to work through the following cognitive developmental stages when teaching mathematical concepts at all grade levels: concrete, pictorial, symbolic, and abstract. Individuals with dyslexia will learn best when provided with concrete manipulatives with which they can work or experiment. These help build memory as well as allowing for revisualization when memory fails. The next stage, pictorial, is one which may be brief, but is essential for beginning the transition away from the concrete. This is where individuals recognize or draw pictures to represent concrete materials without the materials themselves. Symbols, i.e., numerals, plus signs, etc., are introduced when individuals understand the basic concept, thereby making the connection to procedural knowledge. Finally, the abstract stage is where individuals are able to think about concepts and solve problems without the presence of manipulatives, Pictures, and symbols. (Steeves & Tomey, 1998a).6
According to Steeves and Tomey (1998a),7 it is important that the four developmental stages are linked through language for these individuals. There are three kinds of language which allow one to fully integrate mathematical learning. First, is the individual's own language. No matter how imperfect this language is, it is important that the individual discusses, questions, and states what she/he has learned. Second, is the language of the instructor, or standard English, which clarifies the learner's own language, and links to the third language, the language of mathematics. The language of mathematics is not just the vocabulary but the use of sign, symbols, and terms to express mathematical ideas, such as 2 + 4 = 6. Also, language allows the instructor to determine if the learner understands the concept and is not just following steps demonstrated by the instructor to complete a process, even at the concrete stage.
Wright, C. Christina (October 1996). Learning Disabilities in Mathematics. Accessed 9/7/03: http://www.ldonline.org/ld_indepth/math_skills/math1.html
When children enter school, they will gradually learn the format aspects of number ,i.e., adding with exchanging and trading. In the best circumstances, children begin with informal mathematics, usually with manipulatives, and gradually build to the more abstract, less inherently meaningful formal procedures.
Many children do not make this connection and characterize math as a collection of unconnected facts which must be memorized. They don't look for patterns or meaning and can feel puzzled by classmates who seem to learn with so much less effort. In other cases, adults move in prematurely with children who are eager and excited to memorize, teaching them procedures which they can imitate but not understand. While this informal/formal gap is not, strictly speaking, a learning disability, it probably is a factor in a majority of math learning difficulties.
The pace at which children move from informal to formal arithmetic is far more gradual than most educators or parents realize. Even as adult learners we need a considerable chunk of time with the concrete, "real" aspect of a new piece of learning before we move on to making generalizations and other abstractions.
This article and other articles include a focus on math learning problems and perhaps math disabilities that may be brought on by poor instruction. Roughly speaking, the current math curriculum works all right for a majority of students. Since the math curriculum changes only slowly over a period of years, parents are often able to help their children learn math—by teaching in the way they were taught. Similarly, many elementary school teachers teach math in the way they were taught. This "teach the way in which you were taught" breaks down when a student has math learning and/or math doing difficulties.
