According to the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5), learning disorders are among the most frequently diagnosed developmental disorders in childhood. Children experiencing a deficit in one learning domain frequently show deficits in other domains likely due to shared genetic variance.
The definition of a mathematics learning disorder includes well below average mathematical academic performance for age that is not attributable to an intellectual disability (which is defined by IQ below 70) or a predefined discrepancy between IQ and the affected learning domain.[1]
Neurologic in origin, learning disorders impede a person's ability to store, process, and/or produce information. Learning disorders can affect the ability to read, write, speak, or compute mathematics and can impair socialization skills. The central clinical feature of a learning disorder is the lack of normal developmental skill, either cognitive or linguistic.
Mathematical learning disorder (MD) also known as dyscalculia is a term used for a wide range of disorders caused by abnormalities in one or more of the basic psychological processes involved in understanding or use of math. Several manifestations of the disorder may occur throughout the life of the individual. Mathematical learning disorder does not include children who have learning problems caused primarily by (1) visual, hearing, or motor impairments; (2) mental retardation; (3) emotional disturbance; or (4) environmental, cultural, or economic disadvantages.[2, 3, 4]
US clinicians should become familiar with the federal Individuals with Disabilities Education Act (IDEA), which defines learning disorders as "processing disorders that result in a significant discrepancy between potential and acquisition of various academic or language skills."[5, 6] Although this definition has raised several questions, it remains important in current clinical practice. Mathematical learning disorder is among the disabilities that qualify children for special education programs under IDEA.
Assessing the exact incidence of mathematical learning disorder is difficult due to paucity of studies that focus specifically on basic number and arithmetic skills.
Collectively, learning and language disorders comprise a very common set of problems. An estimated 10–20% of children and adolescents have a language disorder, a learning disorder, or both. Reading disorders (RDs) comprise a large portion of this group. An estimated 3–7% of elementary school children have been identified with mathematical disorder comparable to the percentage with reading and spelling disorder.[1] However, children often have more than one disorder; 56% of children with a reading disorder also showed poor mathematics achievement, and 43% of children with a mathematical learning disorder showed poor reading skills.
The estimated incidence may not accurately reflect the presence of the disorder. Some children may have narrow deficits in certain aspects of arithmetic (eg, counting) and perform well in all other aspects. However, standardized tests will still record a poor performance.
Mathematical learning disorder incidence among American children is higher than in Japanese, German, or French children. This higher incidence may be linked to the instructional course design.
Although children with developmental dyscalculia perform more poorly during subtraction activities, there appears to be greater activity in multiple intra-parietal sulcus (IPS) and superior parietal lobule subdivisions in the dorsal posterior parietal cortex as well as in the fusiform gyrus in the ventral occipito-temporal cortex. A study of connectivity analyses revealed hyper-connectivity, rather than reduced connectivity, between the IPS and multiple brain systems including the lateral fronto-parietal and default mode networks thus suggesting the possibility that the IPS and its functional circuits are involved with inappropriate task modulation and hyper-connectivity during addition and subtraction tasks as opposed to the theory of under-engagement and under-connectivity.[7]
Risk factors include very low birth weight due to maternal cigarette (nicotine) smoking, which can contribute to reduced gray matter volume in the intra parietal sulcus.[8]
Long-term prognosis is guarded as numerical skills have been associated with increased risk for unemployment and stress. The lack of numerical literacy skills can interfere with everyday basic living skills: e.g., not being able to make and keep appointments (due to not being able to judge or tell time), problems with paying bills (which can lead to homelessness), and trouble with the use of social media (passwords) and social interactions (not able to remember phone numbers).[8]
Children with learning disorders typically present at primary school age or later. Often, mathematical learning disorder (MD) is associated with reading disorder (RD), although mathematical learning disorder is noticed later because of language's permeating influence in everyday life. Mathematical learning disorder often goes unrecognized until the child begins schooling.
A multitude of developmental pathways converge when children strive to comprehend and apply mathematics in school.[9, 10, 11, 12, 13] Over time, the demands of the mathematics curriculum impose increasing strain on a developing and differentiating nervous system. Levine and associates' 16-subcomponent model helps clarify the causes of problems performing mathematics and helps evaluate mathematical learning disorder.[14] Subcomponents of the model include the following:[15]
See the list below:
Virtually all mathematical procedures involve a body of underlying factual givens. Mathematical facts include the multiplication tables, simple addition and subtraction, and a range of numerical equivalencies.
Early stages of elementary school mathematical learning generally place heavy reliance on rote memory as a child seeks to incorporate an immense volume of mathematical facts. Once these facts are memorized, the child then must engage in convergent retrieval; facts must be recalled precisely on demand.
An elementary school student then must progress to fully automated recall of mathematical facts. For example, while performing an algebra problem, a student is required to recall principals of addition, subtraction, division, and multiplication accurately and in precise detail.
Elementary school students who face difficulty are those who have problems initially memorizing mathematical facts; those with divergent, imprecise patterns of retrieval memory; and those who have difficulty recalling mathematical facts, which slows their ability to count. These students later have difficulty with more sophisticated problem solving, resulting in mathematics underachievement at middle school level.
See the list below:
Mathematics computations laden with fine detail (eg, order of numbers in a problem, precise location of a decimal, appropriate operational signs [+, -]) comprise the heart of a mathematics problem. High attention to detail is needed throughout the operation of mathematics.
The children most likely to face problems with mathematical computations at this level are those who have attention deficits and those who are impulsive and lack self-monitoring.
A student with attention deficit hyperactivity disorder (ADHD) may appear to understand facts, but that student's lack of attention to detail creates poor overall performance.
See the list below:
In addition to mastering mathematical facts, a student must be able to recall specific procedures (eg, mathematical algorithms). These algorithms include the processes involved in multiplication, division, reducing fractions, and regrouping.
A good understanding of their underlying logic enhances recall of such procedures.
At this level of functioning, children with sequencing problems have significant difficulty accessing and applying mathematical algorithms.
See the list below:
With increasing experience and skill, school-aged children should be able to manipulate facts, details, and procedures to solve more complex mathematical problems, a process that requires integrating several facts and procedures in the same problem-solving task.
The act of manipulation requires a substantial amount of thinking-space or active-working memory. For example, solving a problem often requires students to remember numbers and use them later. Students should be able to understand why they are using the numbers and then use them. Students should also be able to manipulate task subcomponents.
Students with limited active-working memory experience considerable difficulty using manipulations.
See the list below:
Mathematics confronts students with a wide range of recurring patterns. These patterns may consist of keywords or phrases that continually emerge from word problems and yield significant hints about the procedures required.
Students often must be able to discard superficial differences and recognize the underlying pattern, a process that creates problems for students with a pattern recognition disability.
See the list below:
Without question, mastery of mathematics requires the acquisition of a rather formidable mathematical vocabulary (eg, denominator, numerator, isosceles, equilateral). Much of this vocabulary is not part of everyday conversation and, hence, must be learned without the assistance of contextual clues.
Children who slowly process words and who are weak in language semantics falter at this level.
See the list below:
The language of mathematics is unique in the sense that a student is expected to draw inferences from word problems expressed in sentences. Keen sentence comprehension and knowledge of mathematics vocabulary are needed to understand explanations from books and instructors.
Children with language disabilities may feel disoriented and confused by verbal instructions and by written assignments and tests.
See the list below:
Much mathematical subject matter is presented in images and in a visual-spatial format. Geometric figures require keen interpretation of differences in shapes, sizes, proportions, quantitative relationships, and measurements.
Students must also be able to correlate language and figures; the terms trapezoid and square should evoke design patterns in students' minds.
Children with weaknesses in visual perception and visual memory may have trouble with these subcomponents of mathematics.
See the list below:
At middle school level, use of logical processes and proportional reasoning increase. Word problems (eg, if...then, either...or) require considerable reasoning and logic. These concepts are also used in other subjects such as chemistry and physics.
Children who lag in acquiring propositional and proportional reasoning skills may be less able to perform direct computation and word problems that demand reasoning. These students may excessively rely on rote memory.
See the list below:
An important part of the reasoning process, and a problem for children lacking this skill, is the ability to estimate answers to problems.
The ability to estimate solutions to a mathematical problem often indicates the child's understanding of the concepts needed to solve the problem.
See the list below:
Understanding concepts forms the basis of several mathematical problems (eg, 2 sides of an equation should be equal, fractions and percentages are frequently equal).
Children with poor conceptualization abilities frequently have difficulty in middle school mathematics; they may be unable to link concepts and have only fragmentary knowledge of applicable mathematics.
See the list below:
Problem-solving skills are complex abilities that require a systematic strategic approach, entailing the following steps:
Identify the question
Discard irrelevant information
Devise possible strategies
Choose the best strategy
Try that strategy
Use alternative strategies, if required
Monitor the entire process
Impulsive children who fail to use this systematic approach and do not self-monitor throughout the process are unlikely to perform the task in a coordinated, executive-functioning manner.
See the list below:
Mathematics is intensely cumulative. A hierarchy of knowledge and skills must be constructed over time. Information learned in lower grades must be retained for future use. Students can appreciate the Pythagorean theorem only to the extent that they recall the definition of a right triangle.
Some children apparently encounter difficulties developing cumulative memory and recall. They may have problems in subjects other than mathematics that also require cumulative recall (eg, science, foreign language).
See the list below:
Children should be able to realize the relevance of mathematics to learning and use in day-to-day life.
Students unable to perceive this relevance may find mathematics alien or irrelevant.
See the list below:
Apprehensions, anxieties, or phobias are common complications of disabilities in mathematics.
These reactions can be caused by any of the above disabilities or may be rooted in fear of repeated humiliation in class.
See the list below:
Some children have natural affinity to mathematics. These children may have strong role models with an affinity for mathematics, or the children themselves have strong conceptualization abilities.
Students with a natural affinity for mathematics may be keenly aware of the subject's cohesion and can perceive mathematics' beauty and elegance.
See the list below:
Facts - Memorization, retrieval memory
Details - Attention, retrieval memory
Procedures - Conceptualization, sequencing procedural recall
Manipulations - Conceptualization, active-working memory
Patterns - Conceptualization, recognition memory
Words - Language, conceptualization, verbal memory
Sentences - Language conceptualization
Images - Visual processing, visual retrieval memory
Logical processes - Reasoning skills, procedural skills
Estimating - Attention (ie, planning, previewing skills), nonverbal and verbal conceptualization
Concepts - Nonverbal and verbal conceptualization
Risk factors include very low birth weight due to maternal cigarette (nicotine) smoking, which can contribute to reduced gray matter volume in the intraparietal sulcus.[8]
Fetal Alcohol Spectrum Disorder (FASD)
Importance of obtaining a birth history cannot be neglected especially due to the high association of low birth weight, especially from maternal cigarette (nicotine) smokine, and mathematics learning disability.[8]
Mathematics assessments play a valuable role in identifying students' strengths and weaknesses and in developing and monitoring instructional practices. The following assessment strategies are the most popular in use today.
The term portfolio refers to collections of students' work that exhibit their efforts, progress, and achievements in single or multiple subjects. In mathematics assessment, a portfolio can be a useful tool to monitor student learning and the effectiveness of instructional programs. In assembling a portfolio, ensure that content a valid representation of curricular goals, content is collected within a time frame, and content represents various situations. Documented analysis of the student's portfolio that incorporates the following points can be used to monitor student progress on a regular basis:
Answer correct or incorrect
Computational skills demonstrated or lacking
Reading errors that may have contributed to the incorrect solution
Syntactical errors made
Strategy used to solve problem
Visual aids (eg, pictures, graphs) used
Criterion-referenced test results demonstrate student knowledge of specific content that is unrelated to peer performance. These tests present a sufficient number of items to measure various aspects of mathematics skills. The tests are conducted within a specified time period (usually 1.5 times that of an average child's performance time) to identify specific skill deficiencies.
Curriculum-based measurement (CBM) is a validated version of curriculum-based assessment. CBM involves ongoing measurement of a student's actual performance in comparison to the school curriculum's planned outcomes. Because CBM uses the school's curriculum as a basis for comparison, CBM provides great help to teachers on a daily basis by evaluating each student's learning rates, by determining what instruction is needed, and by ascertaining the effectiveness of interventions with individual students.
Calculation error analysis, using structured interviews and checklists, rating scales, or both, is an efficient way to identify a student's calculation strategies. Checklists and rating scales can be used to note strategies used during the interview or strategies observed while the student performs a calculation. Checklists can be dichotomous (yes/no) responses or can use Likert scale (ie, sliding scales ranging from never to always). One approach is for the interviewer to give a student a problem and then ask the student to "think out loud" while working on the solution.
Observations provide valuable data, which should be combined with data accumulated via other strategies to assess the overall effectiveness of instructional efforts. Within the instructional context, teachers continually make informal judgments about student progress.
Gathering information about a student's motivation and confidence level during an instructional activity sometimes proves helpful. Students may respond to a brief survey of questions about their confidence level and any difficulties they encounter.
Early remediation of mathematical learning disorder (MD) is crucial to ensure the child's recognition of mathematics' significance not just in the classroom but also in everyday life. Based on the new information available for reading disorders (RDs), new strategies designed for educators to guide and help nonperforming students improve are available. Work is still needed to identify the basic problems with mathematical learning disorder, however there may be a verbal subtype of MD that involves problems with phonological awareness reflected in impairment of counting speed, number processing, and fact recall, however it is unknown at this time if the processing deficit is specific to symbolic magnitudes or nonsymbolic magnitudes involving conceptual processing and the recall of semantic information from memory.[8, 15]
Mathematical learning disorder (MD) management should begin early in a child's educational career. Unfortunately, mathematical learning disorder is usually not recognized early enough or management is delayed until other problems (eg, language disabilities) are addressed.
Many children perceive mathematics as a subject confined strictly to mathematics class and homework. Early remediation of mathematical learning disorder is crucial to ensure the child's recognition of mathematics' significance not just in the classroom but also in everyday life. Based on the new information available for reading disorders (RDs), new strategies designed for educators to guide and help nonperforming students improve are available. Work is still needed to identify the basic problems with mathematical learning disorder, which will help create improved strategies to help children. Meanwhile, the following guidelines are indicated to help children with this pervasive disability.
Remediation demands close collaboration between regular classroom teachers and those involved in remedial support. Many children with underachievement in mathematics are eligible for legally mandated special education services in public schools. Wide differences are observed in service eligibility requirements, and the quality and intensity of services markedly vary between communities. Identifying the disability of each student and addressing it at the individual level is still important. General remediation guidelines are as follows:
Underdeveloped subcomponents
Intervening at the level of the individual subcomponents is essential (see Causes).
A tutor, a regular or resource classroom teacher, and, under certain circumstances, a parent can help the student work on the specific underdeveloped subcomponent. The concept is for the child to work more on the underdeveloped subcomponent than on getting the correct answer. Examples include supervised practice for a student with poor pattern recognition, designed to review word problems and to identify the key words or patterns that suggest a particular procedure. In another example, a child whose automatic recall of mathematics facts is delayed should practice recalling facts under timed conditions.
Whenever possible, exploit a child's developmental strengths and subject area affinities. A good visualizer should study correctly solved problems and make use of diagrams and other graphic material. A highly verbal child should learn mathematics by trying to teach the subject. In some instances, use of educational software can facilitate learning at the level of the deficient subcomponent.
Bypass techniques
Within regular classroom settings, an often desirable teaching method is to circumvent the deficient mathematical task component. This bypass technique enables a child to learn mathematics despite the presence of a deficient subcomponent. Examples include allowing students who are weak at recalling mathematical facts to use calculators when solving word problems.
Time may be used as another bypass strategy. Students with delayed automatization may take an extremely long time to finish a problem. The bypass strategy for these students may consist of giving them more time to complete the problems or expecting them to solve fewer problems.
Teaching real-life mathematics
Children who have too many deficient components or who have deficient curricular abilities require consistently innovative teaching methods.
Sameness analysis and real-life situations are examples of innovative methods that enable children to learn basic mathematics techniques.
Environment
Provide an ideal environment for work, with few distractions and an adequate supply of tools (eg, pencils, erasers, graph paper).
Some children may need a tutor outside the regular classroom to help focus on the child's disability and avoid classroom pressure.
Management of neurodevelopmental dysfunctions
Mathematics performance may be impaired by other neurodevelopmental dysfunctions (eg, attention deficit hyperactivity disorder [ADHD], language disabilities). Treating these respective problems may greatly enhance mathematics skills.
Selected modes of cognitive training may help improve concept formation, problem solving skill, and, most importantly, memory.
Improving curriculum
Research has revealed that, on average, poor mathematics performance in the United States may be linked to a deficient curriculum in comparison to curricula used in other nations.
In-depth analysis of the curriculum, together with incorporation of various suggested new changes, might improve overall national performance in mathematics.
Future research
A growing movement in the field of mathematical learning disorder acknowledges "number sense."[6]
A "phonemes" concept suggests that an understanding of sound and letters helps develop strategies for educators. "Number sense" is a similar concept.
Gersten et al believe that this is a concept of numbers learned in early childhood and may play a crucial role in understanding of mathematics teaching, especially to children with disabilities.[16] Further research is needed prior to development of concrete strategies towards this goal.
Neurodevelopmental or neuropsychological testing can yield valuable information about the underlying dysfunctions that may impede mathematical learning. These dysfunctions include the following:
Attention deficits
Visual-spatial weaknesses
Language disabilities
Memory problems
Poor sequential organization
An education diagnostician or psychoeducational specialist should examine all areas of academic performance. Educational testing of a child with mathematics underachievement should be performed on a 1-to-1 basis. Other academic difficulties (eg, spelling, writing) often lead to mathematics underachievement.
Evaluation of a child's mathematics performance should be calibrated specifically to that child's age and grade level. Identification of specific developmental subcomponents may have significant implications for remediation efforts. Include the following parameters in a standard examination:
Speed and accuracy of factual recall
Appreciation of quantity (ie, quantity in relation to number concepts)
Recall and appreciation of algorithms
Ability to interpret and solve word problems
Level of concept mastery
Quality of attention to detail
Work pace
Child's affect
Student's approach to problem solving
Extent of automatization
At this time, there is no sufficient scientific evidence that supplementation with PUFAs (Omega-3 fatty acids) should be considered for youth with mathematics learning disorder.[17] Studies of mice have been suggestive that inflammation may explain the deleterious impact on spatial learning of PUFA deficiency.[17]
Originally approved by the US Congress in 1975, the Individuals with Disabilities Education Act (IDEA) is an attempt to remedy problems that contribute to the barriers faced by children with disabilities.[18]
IDEA has been updated approximately every 5 years, the latest of which was in 2004. IDEA aims to strengthen academic expectations of, and accountability for, the 5.4 million US children with disabilities and to bridge the too common gap between the regular school curriculum and what these children learn.
Several ideas have become part of the special education vocabulary because of this law, including free appropriate public education (FAPE), individualized education program (IEP), and least restrictive environment (LRE). These concepts have been built into the special education system to insure equal access to education for all students.
The reauthorization of IDEA 2004 states the following purposes:
1A - To ensure that all children with disabilities have available to them a FAPE that emphasizes special education and related services designed to meet their unique needs and prepare them for further education, employment, and independent living
1B - To ensure that the rights of children with disabilities and parents of such children are protected
1C - To assist states, localities, educational service agencies, and federal agencies to provide for the education of all children with disabilities
2 - To assist states in the implementation of a statewide, comprehensive, coordinated, multidisciplinary, interagency system of early intervention services for infants and toddlers with disabilities and their families
3 - To ensure that educators and parents have the necessary tools to improve educational results for children with disabilities by supporting system improvement activities; coordinated research and personnel preparation; coordinated technical assistance, dissemination, and support; and technology development and media services
4 - To assess and ensure the effectiveness of efforts to educate children with disabilities.
With passage of the IDEA amendments, the US government acknowledged that "Disability is a natural part of the human experience and in no way diminishes the right of individuals to participate in or contribute to society. Improving educational results for children with disabilities is an essential element of our national policy of ensuring equality of opportunity, full participation, independent living, and economic self-sufficiency for individuals with disabilities." IDEA strives to increase the involvement of parents and educators in the care of children with disabilities.
For years, schools were required to wait until a child fell considerably behind grade level before being eligible for special education services. Today, with the release of the final regulations of IDEA 2004, school districts are no longer required to follow this "discrepancy model" but are allowed to find other ways to determine when a child needs extra help. This is being implemented throughout the country through a process called Response to Intervention.
Prior to the implementation of IDEA in 1975, approximately 1 million children with disabilities were shut out of schools and hundreds of thousands more were denied appropriate services. Since then, IDEA has changed the lives of children with disabilities.
Many children now learn and achieve at levels previously thought impossible. As a result, and in unprecedented numbers, these children are graduating from high school, going to college, and entering the workforce as productive citizens.
In the past, as many as 90% of children with serious developmental disabilities were housed in state institutions. Today, 3 times as many young people with disabilities are enrolled in colleges or universities; twice as many 20-year-olds with disabilities are working.
Although significant progress has occurred, the status of children with disabilities still falls short of expectations. The following facts reflect this status:
Twice as many children with disabilities drop out of school, compared with children without disabilities.
Dropouts do not return to school, have difficulty finding jobs, and often end up in the criminal justice system.
Girls who drop out often become young unwed mothers at a much higher rate than their peers without disabilities.
Many children with disabilities are excluded from the curriculum and from assessments used with classmates without disabilities, actions that limit their possibilities of excelling and achieving higher standards of performance.
The following nonprofit organizations provide information, referrals to professionals, and contacts to local groups:
Learning Disabilities Association of America (LDA)
(888) 300-6710
eMail: Learning Disabilities Association of America
National Center for Learning Disabilities (NCLD)
(888) 575-7573
(212) 545-7510
International Dyslexia Association (IDA)
(800) ABC-D123
(410) 296-0232
eMail: International Dyslexia Association
Schwab Foundation for Learning
(800) 230-0988
eMail: webmaster@schwablearning.org
Council for Exceptional Children (CEC),
Division for Learning Disabilities (DLD)
(800) 328-0272
IDEA 2004
Council for Learning Disabilities (CLD)
PO Box 40303
Overland Park, KS 66204
(913) 492-2546
LD Online