Mathematics Disorder

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Twelve-year-old Lois is a sixth-grade student whose grades are all A's except in math. Since first grade she has always had great difficulty doing math problems. She is an excellent reader and has developed an exceptional vocabulary, but despite much effort she has not yet been able to learn her multiplication tables. She still uses her fingers to count and makes many errors when she attempts to do simple adding and subtraction. She seems to grasp the idea of multiplying, but is not yet able to multiply two numbers by two numbers. She finds simple division problems very confusing and still makes many mistakes in counting money or in reading the time from a clock. She has a digital watch because she has not yet mastered telling time from a conventional clock face.

Thus far there has been relatively little research to explore the underlying impairments in mathematics disorder. David Geary (1994) has described two distinct types of impairment in children with mathematics disability:

The first involves difficulties in representing arithmetic facts in, or retrieving facts from, long term memory . . . many MD [mathematics disorder] children have problems remembering basic arithmetic facts such as 5 + 9 = 14, even with extensive drilling. . . . The second involves difficulties in executing arithmetic procedures, such as carrying or trading in complex addition, or in executing counting procedures to solve simple addition problems. (pp. 155-156)

Much remains to be learned about the specifics of such impairments in the cognitive skills required for mathematics. There is some evidence that impairments in this domain, like those of dyslexia, are heritable, but the brain mechanisms involved are not yet known.

Disorder of Written Expression

Leo is an eleventh-grade boy who loves to read and gets excellent grades in all his courses in high school except those that involve extensive writing. Since early childhood he has shared with his father an intense interest in the Civil War. With his family he has visited every major Civil War battlefield. He has many books and videos about that war, and collecting Civil War memorabilia is his hobby. A history course that Leo took during the first semester of his junior year began with a unit on the Civil War. During class discussions Leo often contributed information and anecdotes that interested his classmates and impressed his teacher.

Yet Leo failed the first test of the semester, an essay exam requiring paragraphs to explain aspects of the Civil War that had been covered in the textbook and discussed in class. The teacher was annoyed by his overly terse answers and the lack of elaboration to support his responses. He felt that Leo knew the answers better than anyone else in class. He assumed that Leo had not taken the test seriously. Immediately after class the teacher met with Leo and required him to respond orally to each of the questions on the test. Leo's answers were all top-notch. When the teacher asked Leo why he had written such a poor test paper when he knew all the answers so well, Leo answered, "That happens to me all the time. I can know about something and can talk about it with no problem, but I just can't get the ideas written out on the page. All the teachers tell me that my answers are way too short and that I don't elaborate enough. I'm just a really poor writer."

Chapter 4 explains why written expression is much more demanding of executive functions than are talking or reading. The point here is that, like Leo, some children, adolescents, and adults experience extraordinary difficulty in expressing in writing, particularly expository writing, what they think, feel, and know.

After reviewing relevant research, Rosemary Tannock and I (2000) reported that all three of the basic learning disorders occur at rates two to three times higher among children with ADHD than among children in the general population. One of the few studies that looked for all three basic learning disorders in children with ADHD was reported by Susan Mayes and colleagues (2000). In a sample of children referred to a clinic, they found that 27 percent met diagnostic criteria for reading disorder, 31 percent for mathematics disorder, and 65 percent met diagnostic criteria for disorder of written expression. In sum, 70 percent of those children diagnosed with ADHD fully met the diagnostic criteria for at least one of these three learning disorders. Why are individuals with ADHD impaired by these learning disorders at such high rates?

For decades, ADHD and learning disabilities have been seen as completely separate disorders. ADHD was seen essentially as a disruptive be havior disorder that could be diagnosed by observation and treated with stimulant medications. In contrast, specific learning disorders of reading, mathematics, or written expression have generally been perceived as problems "hardwired" into the brain, problems that could be remediated only by employing special education techniques. Currently the boundary between these two categories of impairment is being reassessed.

In 2000 I argued that impairment of working memory plays a critical role both in ADD syndrome and in the cognitive functions that are disrupted in learning disorders related to reading, math, and written expression.

Reading involves holding in mind and integrating initial portions of a word, phrase, sentence, paragraph, chapter, and so forth long enough to connect these with subsequent portions so that connections can be made and various levels of meaning can be comprehended. Connections must be made between letter shapes and phonemes; diverse associations from elements of long-term memory must be quickly sorted out to select what is appropriate to context and to discard what is not. Smooth execution of these multiple linkages clearly involves not only the learning of phonemes and vocabulary, but also ongoing exercise of short-term working memory.

Likewise, most mathematical operations, from the borrowing and carrying of the simplest arithmetic to the intricacies of the most complex calculations for theoretical problem solving, are highly dependent on working memory. Multiple steps must be prioritized and sequenced, and information must be carried from one operation into another. To do arithmetic and mathematics one needs not only knowledge of specific procedures but also effective working memory. The problem solver's ability to transiently hold "on-line" these various numerical facts and relationships while analyzing problems and invoking appropriate learned procedure is another example of the exercise of working memory.

Similarly, working memory plays an essential role in written expression as one selects and weaves together words and verbal images to convey multiple levels of meaning. In writing, one must hold in mind an overall intention for what is to be communicated in the whole of the phrase, sentence, paragraph, essay, report, chapter, book, and so forth, while simultaneously generating the micro units of words and phrases that will eventually constitute the written work being produced. Complex and rapidly shifting interplay of micro and macro intentions is the essence of creating and self-editing that allows one gradually to shift from the glimmer of an idea, through crude approximations of rough draft, to the greater specificity and polish of a final product. ... In addition to many more specific skills, the whole process of written expression involves ongoing and often intensive use of working memory. (pp. 38-39)

H. Lee Swanson and Leilani Saez (2003) also emphasized the central importance of working memory and executive functions in learning disabilities (LD). They reported that "children and adults with LD are inferior to their counterparts on measures of short-term memory in which familiar items such as letters, words, and numbers, and unfamiliar shapes were recalled" (p. 189). They found evidence .. . indicating both the phonological loop [of working memory] and the executive system as sources of deficit for participants with LD. Either one or both of these components play a significant role in predicting complex cognitive activities such as reading comprehension, arithmetic problem solving, and writing, as well as some basic skills (e.g. arithmetic computation). (p. 194)

Taken together, available data suggest that ADHD, language impairments, and specific learning disorders may not be so separate as has been generally assumed. The common element of impaired executive functions may account for a significant amount of the overlap between ADHD and specific learning disorders and between ADHD and some aspects of language impairment, particularly the pragmatic language impairments described in Chapter 4.

Yet specific learning disorders, speech and language disorders, and ADHD are not aspects of one problem. In speech/language disorders and in specific learning disorders, especially reading disorder, there is evidence that some brain functions are impaired in ways that are not found in individuals who have ADHD alone.

In Chapter 1, I suggested that if the brain is imagined as a symphony orchestra, executive functions impaired in ADD syndrome should be compared not to impaired players of the instruments, but to impairments in the conductor. In the overlap of ADD syndrome with impairments of language and learning, it would seem that this comorbidity reflects dysfunction in some of the players of instruments, as well as in the orchestra's conductor.

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