History
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.
Causes
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' 16subcomponent model helps clarify the causes of problems performing mathematics and helps evaluate mathematical learning disorder.^{ [14] }Subcomponents of the model include the following:^{ [15] }
Learning facts
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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.
Understanding details
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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 selfmonitoring.

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.
Mastering procedures
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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.
Using manipulations
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With increasing experience and skill, schoolaged 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 problemsolving task.

The act of manipulation requires a substantial amount of thinkingspace or activeworking 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 activeworking memory experience considerable difficulty using manipulations.
Recognizing patterns
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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.
Relating to words
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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.
Analyzing sentences
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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.
Processing images
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Much mathematical subject matter is presented in images and in a visualspatial 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.
Performing logical processes
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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.
Estimating solutions
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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.
Conceptualizing and linking
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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.
Approaching the problem systematically
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Problemsolving 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 selfmonitor throughout the process are unlikely to perform the task in a coordinated, executivefunctioning manner.
Accumulating abilities
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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).
Applying knowledge
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Children should be able to realize the relevance of mathematics to learning and use in daytoday life.

Students unable to perceive this relevance may find mathematics alien or irrelevant.
Fearing the subject
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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.
Having an affinity for the subject
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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.
Mathematical subcomponents and the principal neurodevelopmental function(s) each requires
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Facts  Memorization, retrieval memory

Details  Attention, retrieval memory

Procedures  Conceptualization, sequencing procedural recall

Manipulations  Conceptualization, activeworking 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