Pediatric Annals

Mathematical Disabilities

Katherine Garnett, EdD; Jeannette E Fleischner, EdD

Abstract

"What's one and one and one and one and one and one and one and one and one?"

"I don't know, " said Alice, "I lost count."

"She can't do addition, " said the Red Queen.

Lewis Carroll

The purpose of this article is to review evidence regarding the characteristics and prevalence of disorders in mathematics learning, to consider the behavioral and emotional implications of math learning disability, and to describe remedial and therapeutic interventions for individuals with intrinsic difficulty in the acquisition and use of arithmetic understanding and skill.

Current findings indicate that approximately 6% of school aged children have serious deficits in arithmetic achievement. 1 It is not clear to what degree this figure represents children with only math disability, unaccompanied by disabilities in reading, writing, spelling or oral language, or to what degree the arithmetic deficiencies of this sizable group of youngsters may be attributed to instructional neglect. Substantial arithmetic achievement deficits have been found in children classified as learning disabled, although reading and adaptive deficits form the primary basis for their referral to special education.1,2 While math disability has been found in isolation from reading and language disability,3 apparently they are more often manifest in concert.

Among students classified as learning disabled, deficits in mathematics appear to be as pervasive as deficits in reading achievement. Despite this consistent finding,1,4 much less remedial attention is given to math disabilities than to reading-related disorders. This relative neglect may reflect general social devaluation of math over reading literacy or the belief that math difficulties affect a more circumscribed arena of life and employment. Simply being "poor" at math may, in fact, have a limited effect on the lives of many normally intelligent people with otherwise welldeveloped language and reading skills. On the other hand, math learning disability may in and of itself be seriously debilitating and, for some individuals, may be part of a constellation of difficulties affecting both academic achievement and socioemotional adjustment. It is possible that such a constellation of academic and social difficulties emanate from shared underlying neurological deficit, from diverse patterns of neurological dysfunction, or may represent a reaction to repeated failure. In any of these cases, effects on school and life adjustment can be profound and thus warrant systematic and sustained professional attention.

OVERVIEW OF MATHEMATICS LEARNING DISABILITY

Acalculia: Acquired Mathematics Disorder in Adults

Children's disabilities in learning mathematics have only recently begun to be explored. As has been true in the study of other facets of learning disabilities, views of developmental math disabilities have been influenced by neurological perspectives on acquired disorders in adults. Although considerable caution is required in drawing analogies between acquired calculation disturbance in adults and arithmetic learning disabilities in children, insights from adult neurology have provided a basis for preliminary investigations with children.

The term acoiculia, designating acquired disorder in calculation ability, was first applied by Henschen in 1919. 5 Since the disorder was manifested in cases of occipital, frontal, parietal, and temporal lobe lesions and since the behavioral symptoms included both combined and singular difficulty in retrieving number words, printing numerals, and carrying out math computation, Henschen concluded that probably there exists a distinct cortical network for arithmetic function and that, additionally, the integrity of several cortical areas is needed for calculation.

Later efforts to map the cortical pathology of acalculia have yielded considerable variation and disagreement regarding functional and anatomical clusters. Groupings have been proposed by Luria,6 Hecaen,7 and Cohn.8

Luria proposed four varieties of arithmetic disturbance: 1) Primary acalculia - disintegrated visual spatial synthesis (associated with lesions in the parietooccipital region); 2) Acoustic aphasie acalculia -…

"What's one and one and one and one and one and one and one and one and one?"

"I don't know, " said Alice, "I lost count."

"She can't do addition, " said the Red Queen.

Lewis Carroll

The purpose of this article is to review evidence regarding the characteristics and prevalence of disorders in mathematics learning, to consider the behavioral and emotional implications of math learning disability, and to describe remedial and therapeutic interventions for individuals with intrinsic difficulty in the acquisition and use of arithmetic understanding and skill.

Current findings indicate that approximately 6% of school aged children have serious deficits in arithmetic achievement. 1 It is not clear to what degree this figure represents children with only math disability, unaccompanied by disabilities in reading, writing, spelling or oral language, or to what degree the arithmetic deficiencies of this sizable group of youngsters may be attributed to instructional neglect. Substantial arithmetic achievement deficits have been found in children classified as learning disabled, although reading and adaptive deficits form the primary basis for their referral to special education.1,2 While math disability has been found in isolation from reading and language disability,3 apparently they are more often manifest in concert.

Among students classified as learning disabled, deficits in mathematics appear to be as pervasive as deficits in reading achievement. Despite this consistent finding,1,4 much less remedial attention is given to math disabilities than to reading-related disorders. This relative neglect may reflect general social devaluation of math over reading literacy or the belief that math difficulties affect a more circumscribed arena of life and employment. Simply being "poor" at math may, in fact, have a limited effect on the lives of many normally intelligent people with otherwise welldeveloped language and reading skills. On the other hand, math learning disability may in and of itself be seriously debilitating and, for some individuals, may be part of a constellation of difficulties affecting both academic achievement and socioemotional adjustment. It is possible that such a constellation of academic and social difficulties emanate from shared underlying neurological deficit, from diverse patterns of neurological dysfunction, or may represent a reaction to repeated failure. In any of these cases, effects on school and life adjustment can be profound and thus warrant systematic and sustained professional attention.

OVERVIEW OF MATHEMATICS LEARNING DISABILITY

Acalculia: Acquired Mathematics Disorder in Adults

Children's disabilities in learning mathematics have only recently begun to be explored. As has been true in the study of other facets of learning disabilities, views of developmental math disabilities have been influenced by neurological perspectives on acquired disorders in adults. Although considerable caution is required in drawing analogies between acquired calculation disturbance in adults and arithmetic learning disabilities in children, insights from adult neurology have provided a basis for preliminary investigations with children.

The term acoiculia, designating acquired disorder in calculation ability, was first applied by Henschen in 1919. 5 Since the disorder was manifested in cases of occipital, frontal, parietal, and temporal lobe lesions and since the behavioral symptoms included both combined and singular difficulty in retrieving number words, printing numerals, and carrying out math computation, Henschen concluded that probably there exists a distinct cortical network for arithmetic function and that, additionally, the integrity of several cortical areas is needed for calculation.

Later efforts to map the cortical pathology of acalculia have yielded considerable variation and disagreement regarding functional and anatomical clusters. Groupings have been proposed by Luria,6 Hecaen,7 and Cohn.8

Luria proposed four varieties of arithmetic disturbance: 1) Primary acalculia - disintegrated visual spatial synthesis (associated with lesions in the parietooccipital region); 2) Acoustic aphasie acalculia - inability to calculate aloud (associated with lesions of the left temporal region); 3) Motor aphasie acalculia - impaired internal speech; 4) Frontal lobe acalculia - impaired performance of serial arithmetic operations, such as counting backwards.6 While Luria has emphasized the centrality of visual-spatial disturbance in acalculia, others have viewed primary acalculia (anarithmetia) as neither specifically visuospatial nor specifically verbal and, additionally, have noted the relatively common occurrence of acalculia within an "aphaseological context."9,10

The taxonomy first proposed by Hecaen7 has been supported by Benson and Denckla9 and Benson and Weir11: 1) Aphasie acalculia - calculation errors based on inability to handle numbers as words (attributed to dominant hemisphere lesions); 2) Visuo-spatial acalculia - calculation errors produced by an inability to align problems or to sustain place holding values (attributed to minor hemisphere lesions); 3) Anarithmetia - primary loss of calculation ability, inability in retrieval and/or manipulation of learned arithmetic values (associated with widespread pathology).

Cohn,8 on the other hand, distinguished two major categories of acalculia: one involving disturbance in arithmetic ordering, the other associated with disturbance in memory processes. According to Cohn, manifestations of primary ordering deficit include misalignment of vertical and horizontal number sequences and transposition of numerals. Symptoms of prominent memory deficit include confusion with number carrying, failure to follow the operation sign, and lack of memory for multiplication facts.

The study of adult acalculia is made complex by the diversity of calculating deficiencies and by the diversity of concomitant language disorders, graphomotor disturbances and other signs of neurological dysfunction. Interestingly, adult acalculia, in any of its manifestations attributed to whichever cortical site(s), is not often found in isolation from other neurological pathology,8 although a cohesive syndrome of symptoms has yet to be established. While Gerstmann's Syndrome, a proposed cluster of acalculia, finger agnosia, right-left disorientation and agraphia, was widely touted through the early 1960s, substantial evidence invalidates it as a bona fide syndrome in either adults or children.12'13 Most "Gerstmann- like" patients manifest only some of the characteristic Gerstmann signs, often accompanied by other symptoms of neurological disorder.

Taxonomies and clinical findings from adult neurology, while suggestive, are not and cannot be sufficient for understanding developing children's disabilities in acquiring arithmetic skills for the first time. Instructional as well as developmental factors may well overrule any erstwhile neurological similarities or parallels with acalculic adults.

Mathematics Disability in Children

Given the influence of neurological perspectives in learning disabilities, some investigators use the term developmental dyscalculia to designate children's intrinsic difficulty in learning mathematics. As in adult acalculia, taxonomies have been proposed for developmental dyscalculia. Kosc14 has proffered six sorts of dyscalculia: verbal (oral), lexical (reading), graphical (writing), pragnostic (mathematical manipulation of real/pictured objects), ideognostical (math concepts, relationships and mental calculation), operational (procedures or applying appropriate algorithms). These categories highlight some children's particular difficulty in grasping relationships and developing underlying concepts and other children's difficulty in developing procedural skill. While interesting, it is important to note that these categories have yet to be validated.

Badian4 proposed a classification model for developmental dyscalculia, roughly following Hecaen's outline: number alexia/agraphia, spatial dyscalculia, and anarithmetia. From the results of an empirical study, Badian added a fourth subtype - attentional-sequential. Forty-two percent of the dyscalculic children fell into this attentional-sequential group, 24% into the spatial subtype, 14% anarithmetic, 20% mixed disorders, and none manifested number alexia/agraphia. The largest subgroup, the attentional-sequential, rarely made spatial errors and generally knew how to perform the arithmetic processes presented, although they did not do so correctly. They frequently omitted one of the figures in a column, forgot to combine a carried figure, neglected to include decimal points and dollar signs, did not switch operations when the sign changed, and, most prominently, did not readily recall basic addition/subtraction number facts and multiplication tables.

Cohn15'16 considers developmental dyscalcuHa to be one manifestation of a neurological disorganization syndrome which also underlies delayed acquisition of language. He conceives arithmetic ability as a component of symbolic operations, thus viewing children's dyscalculia as intimately connected with learning disability in other symbolic operations such as reading and writing. Cohn's16 list of manifestations includes malformation in writing numerals, failure to note operational sign or to use lines to organize, failure to remember multiplication facts, confusion with "carrying," and difficulty sequencing procedures and/or numerals.

Strang and Rourke,17 reporting on a series of investigations, provide evidence for different types of math disabilities based on relative strength of arithmetic as compared to reading and spelling performance. They propose that different neuropsychological strengths and weaknesses underlie the same level of poor math achievement in children with differing academic patterns and that these several patterns require differentiated remediation. This point gains importance when considering that some children may suffer extreme difficulty in acquiring computation skills, although they experience no notable difficulty with mathematical concepts or logic, whereas others may master computation with relative ease, only to be unable to apply these skills in solving problems.18

While it is tempting to explain math learning disabilities primarily on neurological grounds, a recent research report from the Collaborative Perinatal Project demonstrates that, for the 1000 learning disabled 7-year-olds in their 35,000 child sample, "the primary causal factors reside not in the child's biomedicai history, but in the child's environment, the social context of development."19 The powerful findings from this large-scale longitudinal research point medicine, education, and psychology to a reconsideration of suppositions regarding the primacy of perinatal distress factors in large numbers of cognitively impaired children. Apparently, such factors by themselves have minimal effects when children are in a responsive and supportive social environment. Conversely, significant handicap results in the presence of inadequate environmental support for children with and without perinatal difficulties. This finding enjoins many sectors of society - medicine, welfare, education - to take much more substantial account of the contributions of socioeconomic status, the early caretaking environment and the educational support afforded to at-risk infants and young children. Results from the Collaborative Perinatal Project caution us not to fix on neurological status as necessarily the primary etiological factor in large numbers of children who manifest both math and other learning disorders.

Developmental Aspects of Math Learning Disability

A good deal of interest in developmental aspects of math learning is reflected in current literature,20'22 However, this interest has not yet resulted in clear early indicators of specific disability in learning mathematics. Manifestations of math learning disability undoubtedly are present during the preschool years, although most go undiagnosed until the end of the primary grades, or later. In addition to the genera! cogniti ve- developmental guides provided under a Piagetian framework (eg, the need to establish rudimentary seriation, classification, one-to-one correspondence, and conservation of quantity), other suggestive evidence exists, offering direction for muchneeded future investigation. For example, Kellogg,23 among others, found a relationship between math learning problems and children's figure drawings; young children who misrepresented number of body parts, especially number of fingers and omission of the nose, were found 6 years later to have significant math achievement deficits. Cohn15 speculates that young children's particular difficulty synchronizing pointing and counting is an early symptom of math disability. In a study of normal and at-risk kindergarteners, Katz24 found that the at-risk group could not count as far, had difficulty with ordination (first, second, third, etc), and could not solve simple word problems that were easy for the normally developing children. These amounted to clear qualitative differences between the two groups in basic counting skills which underlie early math learning.

Mathematics is a language, a means of representing reality through a symbol system composed of ideographs (numbers and signs) which convey concepts and ideas. Disabilities of language functioning can significantly impede development of math understanding, perhaps especially in the early years when vocabulary, verbal counting patterns, and mathematical "syntax" are being introduced.25 Verbal factors have been found to be significant in math achievement26 and may well influence performance beyond the earliest school years, as both mathematical applications and more complex written computation require increased verbal monitoring.

During the elementary school years, the most apparent manifestation of math learning disability is likely to be failure to master basic arithmetic facts. Children are expected to develop understanding of these more simple number combinations (eg, 4+6, 9 - 3, 5x7) and, with practice, to easily recall them for use in more complex computation and problem-solving. In a study designed to identify differences in basic fact recall of normally achieving and learning disabled children, Fleischner, Garnett and Shepherd27 found that speed rather than accuracy distinguished the two groups. The slower computation of the learning disabled children appeared to be linked to their use of immature strategies: they were more apt to count one number at a time, using their fingers, marks, or number lines, counting aloud or subvocally. It is important that their accuracy in calculating basic facts (defined as percent correct of number attempted) was equivalent to that of the normally achieving children. This indicates that, as a group, lack of mastery of these basic number combinations is not fundamentally a conceptual problem in children classified as learning disabled.

Proficiency in elementary arithmetic requires more than mastery of basic facts, however, and math disability in the elementary years is not confined to inefficient basic fact computation, despite the widespread and persistent nature of that particular manifestation. Learning disabled students frequently have great difficulty grasping and reliably using procedures for regrouping and renaming (borrowing and carrying). While in some cases they seem not to understand concepts of place value implicit in our decimal system, in many others their conceptual grasp seems adequate, but they are unreliable in performing the appropriate sequence of procedures. This sort of procedural unreliability may also be at the root of difficulties mastering more complex written computation, such as long division and multidigit multiplication.

Underlying visual-spatial and/or language disorders undoubtedly play a role in the difficulty some students have in carrying out complex computations, as well as in learning to tell time, to use graphs, grasp fractions and understand geometry. Learning disabled students' performance in these many subareas of the elementary curriculum has not received any systematic investigation. Additionally, while overlays such as math anxiety, general performance anxiety, and level of selfesteem have received much attention in the literature of education and psychology, there has been virtually no investigation into their linkage with bona fide math learning disability.

Investigation of arithmetic "story problem" performance of learning disabled fifth and sixth graders revealed a few students with extreme conceptual difficulty, despite adequate computation skills.28 Nuzum found that this subgroup could not develop, act on, and monitor the effectiveness of a plan of action necessary to solve story problems, although they could easily read these and could do the computations in isolation. Solving story problems, in fact problem-solving in general, appears to require deploying cognitive strategies which proceed from analyzing available information and determining what is needed, to selecting appropriate actions in sequence, monitoring performance, and, finally, comparing with an estimate to check for general accuracy and logic. Over a series of different, although generally nonmathematical tasks, numerous recent studies have characterized learning disabled students as lacking in the spontaneous and/or consistent use of these sorts of cognitive strategies,29,30 Again, acrossa variety of domains, training studies have consistently shown that, with sufficient explicit instruction, learning disabled students are amenable to activating such strategies and thereby performing specific tasks successfully.18,31

Although learning disabilities teachers have reported that two out of three of their intermediate and secondary students have deficits in mathematics,32 very little is known about the effect of math learning disability on higher order math achievement. Often, students who do poorly at the arithmetic computation level are counseled out of higher math courses such as algebra, geometry, and calculus and put into "remedial" math, or are counseled into modified curricula where the pace of instruction is much slower than for other students of similar IQ. These sorts of intensive supports can be reasonable and appropriate for some learning disabled secondary and college students, although "remedial" courses are often designed for those who "missed" certain instruction, rather than for those with inordinate difficulty, and thus are often misfocused for the particular needs of the learning disabled. On the other hand, for some secondary and college students with learning disabilities, higher mathematics may actually be easier than elementary arithmetic (Einstein was apparently one such), since it relies more on a specific "numerical" factor and less on the automaticity of habitual routines and procedural reliability. Of related interest, this difference in the weighting of factors in higher math performance may also account for the male/female differential found after elementary arithmetic.33 The possibility of strong potential for "higher" math, despite persisting problems in "lower" arithmetic, may be important to search out in determining the appropriate mathematics course of study for a given individual. It suggests that timed, entirely written placement exams may yield seriously misleading results for some learning disabled adolescents and young adults. Probing of math understanding is called for, including circling incorrectly computed problems to allow students to recalculate and demonstrate the actual degree of their understanding.

In summary, mathematics learning disabilities may be manifested in a number of ways, which vary with age and educational level of the individual. Children in the preschool years may show early signs of math learning disability through failure to learn counting sequences and ordination, and difficulty synchronizing touch counting; they may show distinctive patterns of deviance in human figure drawings, omitting or misrepresenting numbers of features of the body, and may have difficulty with language aspects of arithmetic. During the elementary school years, math disabled students may fail to master basic number facts for ready recall, are likely to be highly unreliable in carrying out the procedures of written computation and may be unable to solve simple story problems, even when they can read them easily and can do the computations separately. Frequently, learning disabled adolescents and young adults are counseled not to attempt the math courses offered to students of similar intellectual ability, or they may require extraordinary special help to master the skills and concepts covered in these courses. On the other hand, some students who demonstrate significant learning disability in the domain of elementary arithmetic may have adequate or better potential for learning higher mathematics, while their arithmetic computation may or may not become significantly more reliable.

SOCIAL AND EMOTIONAL IMPLICATIONS

Two aspects of social and emotional adjustment must be considered in concert with math learning disabilities. First, there is reason to believe that some individuals with intrinsic math disabilities have disorders in perception of nonverbal information and in social interactions. This constellation of disorders is seen as having far-reaching academic, vocational, and life adjustment consequences. Secondly, anxiety can play a serious and debilitating role in depressing math performance, affecting both those with otherwise normal propensity for math learning and those with specific impediments to mastery of basic functional mathematics.

Certainly, learning disabilities can exist as isolated phenomena without concomitant psychopathe logy. Porter and Rourke34 found that about half of their clinic-referred sample of learning disabled children showed clear signs of ". . . balanced and well-adjusted socioemotional functioning." Specific math learning disability can exist independent of reading disability, writing disability, attentional deficit, or socioemotional involvement. However, just as a majority of learning disabled children exhibit combined reading, arithmetic, and adaptive deficits,1 so a substantial number of learning disabled individuals also show signs of socioemotional disorders.

Neither the sources nor the interactive effects of socioemotional disorders which sometimes accompany math disabilities have been clearly established, although speculation has abounded since Johnson and Myklebust's35 early work. Strang and Rourke17 provide some evidence for the existence of a subgroup of math learning disabled children who also exhibit significant socioemotional disorder which these authors attribute to a particular configuration of neuropsychological deficits. This subgroup was first identified by their particular academic pattern: arithmetic performance markedly deficient, in contrast to reading and spelling scores which were at least average on the Wide Range Achievement Test (WRAT). Subsequent extensive neuropsychological assessment revealed impairments in the perception, analysis, organization, and synthesis of nonverbal information (introduced either visually or through touch) as well as deficits in psychomotor output and nonverbal problem-solving, leading the authors EO name this subgroup "nonverbal perceptual-organizational-output disability" (NPOOD). Accompanying these underlying weaknesses were strong language -re Ia ted skills, particularly of the more "automatic" or rote sort.

From both accumulated clinical and limited empirical evidence, Strang and Rourke17 describe NPOOD children as having poor graphomotor skills, as awkward and often clumsy, overly disconcerted by novel social situations or minor alterations in familiar situations, maladept in social interactions, often friendless (particularly as regards age peers), and overly talkative. They propose that these youngsters do not attend to or understand many nuances of gesture or signals provided by the nonverbal context of communication. Additionally, NPOOD children have not developed an appropriate repertoire of their own nonverbal behaviors, including posture, facial and body expressions. Their language skills, strong in some ways, are also often overly fact-laden, cliche -ridden, elaborate, and not sufficiently to the point. The authors surmise that some of the talkativeness serves to alleviate anxiety in confusing situations. It also may serve as the only reasonably reliable way these children have to gather new information or direct their own actions, since their visual information processing capabilities are limited. Additionally, despite the fact that their over- talkativeness may be alienating to many listeners, this language stream may provide their major means of maintaining interpersonal contact. The importance of exploring the validity of this NPOOD description is in the significant and distinctly different treatment implications for such a subgroup.

Strang and Rourke17 are not alone in proposing that some youngsters' arithmetic and social deficiencies may be intimately linked by a common underlying nonverbal learning disorder. Johnson and Myklebust35 also have proposed such a connection, emphasi2ing these children's poor visual-spatial organization, their body image impairment, right-left confusion, poor sense of direction, social difficulties and immaturity. Importantly, Johnson and Myklebust caution that a child with such a constellation of difficulties is easily misjudged, even by experienced helping professionals: aspects of a youngster's nonverbal deficit, especially an intrinsic social imperception, are easily mistaken for emotional disturbance, thereby initiating inappropriate treatment modes and methods for both the child and family.

In sum, a sizable body of clinical evidence describes a subgroup of math disabled children whose math disorder is accompanied by socioemotional dysfunctioning. It has been proposed that these difficulties share a common base in a nonverbal disorder. While there has been minimal direct validation of this subgroup, it remains a viable and potentially important proposal, worthy of further empirical pursuit. It seems clear that such a description will not cover the large number of cases of children with math performance deficits, but may illuminate a small, overlooked, and seriously impaired subgroup.

Anxiety has long been associated with substantial reduction in math performance, particularly in mental calculation. Among those referred to psychiatrists, psychologists and special educators for math learning difficulties are a group of individuals, young and older, who suffer marked anxiety when confronted with reasonably simple arithmetic problems. Consider the case of a self-supporting woman of thirty who works as a hair dresser. On those days that she works, she refrains from smoking until about 3:30, at which time she also begins to talk about needing to balance receipts against cash in her drawer. Her work day ends at 6:00 and, as this hour approaches, her increasing consternation provokes smoking half of a pack of cigarettes, eating constantly, and repeatedly switching the conversational topic to whether she will be able to tally cash on hand to equal her addition of receipts. This young woman was classified as learning disabled during her public school years, and received extensive special education which enabled her to overcome reading disabilities. As is the case with so many learning disabled students, math was largely neglected in her education's overly-narrow focus on her significant reading needs; she was simply never provided with intensive instruction in math. Even the simple computations required in her daily exchanges provoke repeated bouts with anxiety. Because of the general neglect accorded math difficulties, it is not at all clear how varying degrees of math anxiety interact with differing sorts of math learning disability. It seems likely that there is a tortured cycle for some individuals, when poor math skills lead to realistic feelings of inadequacy, and anxiety reactions occur which further diminish math learning and performance.

TREATMENT OF MATH LEARNING DISABILITIES

As is the case with individuals who show signs of other forms of learning disability, those with math learning disabilities frequently require assistance from members of several professions if their problems are to be overcome. A critical 6rst step in devising an appropriate treatment plan is a comprehensive evaluation. At minimum, such evaluation includes psychological appraisal to determine cognitive ability level (IQ); medical evaluation to determine neurological status and to explore psychosocial adjustment; and educational evaluation to determine achievement levels, error patterns, and useful teaching tactics. Such comprehensive assessment may determine that remedial teaching alone is needed. For many, highly systematic and thorough remedial math teaching can result in significant progress in areas of particular math difficulty.27 For others, one-to-one special teaching is called for. Such math tutoring can provide powerful academic and emotional support, although it may also need to make use of nonstandard, carefully tailored teaching techniques. Still others with math learning disabilities may need a multimodality treatment plan, including some combination of parental counseling, conferencing and coordinating with school personnel, specific social skills training, and appropriate psychotherapy.

Principles of Remedial Instruction

Special educators often rely on three principles in developing remedial plans for students with math learning disabilities: 1) clarify underlying concepts by illustrating them in a variety of ways; 2 ) ensure reliable application of sequential computational steps; and 3) teach the effective use of aids, such as number lines and calculators.

Arithmetic operations are based on assumptions about the effect of combining and partitioning objects. Thus, addition and multiplication involve combining; subtraction, division, and basic fractions involve separating or partitioning. For the rationale behind computation to be clear, these concepts must be illustrated through concrete representations. Many young children develop intuitive understanding of these concepts through the many concrete illustrations available in life experiences; many with learning disabilities need special help developing a solid conceptual base. Therefore, special educators commonly use a variety of manipulative materials, such as blocks, abacuses or Cuisenaire rods to develop the logical underpinnings of computational procedures. The use of a variety of manipulatives, accompanied by or perhaps preceded by succinct verbal description, may be critically important for students with visual-spatial disorders, including those similar to Strang and Rourke's17 NPOOD group.

All children need to establish rote counting sequences as an early foundation skill; some math disabled children seem not to have caught on to the pattern and system inherent in continuous counting. This includes the extending of counting sequences decade by decade, counting backwards, skip counting, counting by fives and by tens.

Proficiency in arithmetic computation depends on efficient recall of number facts and skill in sequentially applying computational steps, as well as on underlying concepts and counting skills. Remedial math instruction must emphasize mastery of basic facts, for these are the building blocks of complex computation. Several methods have proven useful in assuring that basic facts can be recalled readily; all share the common element of sufficient practice using highly motivational activities. Recent experience with computer assisted instruction suggests that it may be an especially powerful medium for the needed drill and practice, since students very willingly work for long periods on computer-based drill activities. Self-record keeping, whether a feature of a computer program or a simple instructional structure, is important, permitting students to notice and monitor progress and providing incentive for further effort.

In order to learn the steps in complex computational procedures, some students require visual prompts. For instance, using a highlighter on a multidigit multiplication example to emphasize where to record the different steps can help clarify spatial-visual aspects. Other students may benefit from direct verbal instruction in the mechanics of arithmetic. Strang and Rourke17 suggest that verbal rules and routines should be presented and practiced until they serve to guide computational behavior. Verbal mnemonics have proven helpful (eg, Diane Makes Super Chocolate Brownies ... for Divide, Multiply, Subtract, Compare, Bring down).

While computational proficiency is an important goal of the mathematics curriculum, it should not be stressed at the expense of instruction in higher mathematics for students who are intellectually able to master higher math. As was noted earlier, math placement in high school and college often is predicated on inappropriate tests of computational ability. Calculators can serve as important aids for certain individuals, if they are taught to use them effectively. "Talking" calculators, now available to the average consumer, are proving invaluable for those who frequently enter 69 in lieu of 96. Calculators, though, are not a panacea as they are helpfiil only when an individual has a good grasp of the computational operation to be performed.

CONCLUSION

When the Red Queen remarked that Alice couldn't do addition, she showed little appreciation for the complexities of learning mathematics. Although our society is rapidly becoming a technocracy, many citizens share the Red Queens apparent perception that math skills are just a matter of learning to count. The intent of this article was to point out the interdependencies among neurodevelopmental and socioemotional status, cognitive ability, and environmental foctors in learning mathematics and in math teaming disabilities. Effective treatment of math learning problems requires cooperation among special educators, psychologists and physicians; multimodality treatment can significantly support individuals with math learning disabilities in becoming better achievers in school, better adjusted people, and better able to apply mathematics to the everyday problem-solving of their lives.

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