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Oneof the causes of this cycle of failure, fear, and low achievement in mathematics, for many children, is known as dyscalculia. However, not all learning difficulties in mathematics are due to dyscalculia. Mathematics learning difficulties fall on a continuum of difficulties, problems, or disabilities. Students may have difficulty in learning Algebra with intact numeracy skills. That difficulty is not due to dyscalculia. Similarly, a student may have difficulty in learning geometry with intact numeracy and algebraic skills. However, some difficulties in algebra may be due to dyscalculia, particularly when numeracy is involved, and a student has poor numeracy skills. Dyscalculia, a developmental condition, reflects difficulties in some of the associational skills basic to number conceptualization, number relationships (arithmetic facts), and understanding and executing numerical operations. It affects early numeracy acquisition and then wherever[3] numerical manipulations are involved.

Dyscalculia typically occurs in cases of normal intelligence, although scores on some intelligence subtests related to spatial orientation/space organization may be low. Some numeracy skills might be present, but the levels are significantly low, particularly mastery of arithmetic facts and the ability to estimate the outcome of numerical operations. Dyscalculics calculate haltingly and simple arithmetical errors are often present in their work. A fact might be known in one context and not present in another. Most of the calculations are based on counting (finger reckoning, number line, hash-marks, counting discrete objects, etc.) and often with the help of other devices, such as calculators, number tables, and charts. Frequently, the dyscalculic will guess answers with superficial number relationships. Their number attack strategies are hardly based on decomposition/recomposition of number. They lack effective, efficient, and generalizable strategies.

Dyscalculics may know some facts, but they lack the understanding of the fundamental concepts of subitizing, decomposition/recomposition, properties of the BaseTen-System (making tens, teens numbers, next tens, place value, meaningful estimation, reasons behind procedures, etc.), properties of operations and the concept of equality. Therefore, they have problems in extending their limited knowledge of facts, place value, and meaningful numbersense.

Dyscalculic cases often have poor memory (short-, working, and/or long-term), and indeed do for reading and numeracy material. Their learning strategies are not well-established. They use inefficient strategies that do not help them to receive, process, retain, recall, and produce information properly. Poor teaching exacerbates it. For example, learning arithmetic facts by rote or by counting places heavy load on working memory and since, by this process, the facts are placed in the long-term memory in isolation (because of memorizing isolated facts by flash cards without strategies), their retention (because few, poor, no connections are made between facts and concepts) is difficult, and the recall of unconnected information is difficult. Since they have fewer facts mastered, they have difficulty making connections, extensions, and applications.

This comorbidity between dyslexia and dyscalculia indicates that there are fundamental components in both: phonemic awarenessand decomposition/recomposition, respectively. On the other hand, high comorbidity rates between reading disorder (RD) and mathematics disorder (MD) indicate that, although the cognitive core deficits underlying these disorders are distinct, additional domain-general risk factors might be shared between the disorders. Three domain-general cognitive abilities processing speed, temporal processing, and working memory are studied in RD and MD literature. Since attention problems frequently co-occur with learning disorders, the three factors, which are known to be associated with attention problems, account for the comorbidity between these disorders. However, the attention problems observed in the case of MD, some of them are secondary, in the sense, that they might be the byproduct of consistent failure in mathematics rather than the causative factors.

After controlling for attention, associations with RD and MD differed: Although deficits in verbal memory were associated with both RD and MD, reduced processing speed was related to RD, but not with MD; and the association with RD was restricted to processing speed for familiar nameable symbols. In contrast, impairments in temporal processing and visuospatial memory were associated with MD, but not RD. Visuospatial memory is essential for visual clustering, decomposition/recomposition, and therefore with development of sight facts.

A dyscalculic may not show gross defects on a neurological level, however, in some cases, there may be some non-focal abnormalities in brain functioning. On neuropsychological examination, there may be some deficits in some aspects of executive functioning. There may be some visuospatial-perceptual integration issues. Visual perception integration, working memory, and spatial orientation/space organization are highly correlated with mathematics learning and their poor development becomes a factor in the incidence of dyscalculia. As in all learning disability cases, children with mathematics difficulties, whether dyscalculia or other mathematics difficulties, feel inadequate, stupid, and guilty of their disability and their repeated failures.

The specific elements of disability associated with dyscalculia appear to be inefficiencies in the development of number concept: (a) associating and integrating (i.e., visual representation of a collection of objects, orthographic representation, and the oral representation of quantity); (b) decomposition/recomposition of number in developing arithmetic facts and operations; (c) developing and mastering arithmetic facts; and (d) estimating the outcomes of numerical operations. These deficiencies and inefficiencies grow into problems in mastering numeracy.

The neurobiological revelations in recent neurological research are inspiring new treatments to a variety of disorders. Given the importance of neuroplasticity in very young children, specialists now advise the opposite of a wait-and-see approach. The brain findings affirm the idea that we want to get involved in helping children as early as we can. Early identification and assessment of number concept are essential to prevent numeracy failure in young children and avoiding future mathematics difficulties. Students without adequate mastery of sight facts[6] and decomposition/recompositionearly, continue to demonstrate poor numbersense and numeracy skills, even into the middle grades and high school. There is a strong predictive validity of later mathematics achievement in mastering the components of early number concept: decomposition/recomposition and sight facts. Children continue to use the number concept and decomposition/recomposition whenever they encounter numeracy problems. For example, when students encounter work on fractions, integers and rational numbers, they continue to need and use them for new number relationships.

The following principles of remediation provide a framework to build effective interventions to use and practice. Several principles are restatements of principles common to all efficient and effective learning processes.

Another important principle is: The student should get comprehensible, equitable input from instruction. It should have mathematical agency. Instruction is effective when it creates positive mathematics identity for the student.

The learning difficulties of dyscalculic children necessitate close adherence to effective general learning principles, effective use of time specific to task and focused concept and skill specific strategies.

These frameworks give rise to some working principles for remedial instruction. However, we should not forget that the incidence of acquired dyscalculia can be drastically reduced if regular classroom teaching also follows these principles.

Teachers learn to violate general learning principles because normal children have such powers of independent learning that convenient violation of the principles do not handicap a child for long. And gifted and talented children may not even adhere to any general learning principles, because these children learn despite us.

When we adhere to important general principles and learning strategies specific to dyscalculic children, then they sometimes make more progress in one session of well-planned, focused, direct remediation than they previously made in a year or more of haphazard remediation. This is because, in later case, violations of learning principles produced negative learning, confusion, and a sense of failure. Remediation that does not help is likely to be harmful.

The same principles work in teaching children numeracy and other related concepts. Learning sequential counting, one-to-one correspondence, and writing numbers in isolation are useful to an extent. But an instruction that focuses on integration of visual clustering (expanded subitizing), building sight facts, and decomposition/recomposition is even more important and productive. An organized, early, intensive supervised practice in the integration of these components develops effective number concept (numberness) and then aids in the optimal development of numbersense and numeracy.

In teaching number, most teachers help children acquire the number concept by connecting the phoneme (sound s-e-v-e-n, as they count) with graphme (symbol 7, looking at the number and writing it). But that is not enough because acquiring numberness is more than learning to read a number, write a number, or even count. It is integrating, the cluster (quantity), phoneme (the name), grapheme (the shape) associated with the number. When numerosity and oral representation are learnt before the writing of numbers, children develop number concept faster. The judicious integration of the two expedites the process.

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