What are the potential problems that can arise with individualized instruction and remediation? Apart from the practical problems arising from inadequate resources, there can be problems with the methods that are used. In particular, there may be gaps in the selection of arithmetical components for remediation. Indeed, in our present state of knowledge there must be, since we do not yet have a full understanding of the different components of arithmetic; the relationships between them; and the nature and causes of individual differences in them. Although such topics have been studied for many years, our understanding of them still has limitations, due to the number (considerably more than seems to be the case for reading) and complexity of the components; the difficulties and controversies involved in defining successful acquisition of numeracy (Brown, 1999; Baroody, 2003; Cowan, 2003); and the fact that some important methods of studying the issues, such as functional brain imaging, have only recently become available. Nevertheless, there has been sufficient knowledge in the area for quite some time to permit successful work in the area.
Moreover, appropriate individualized instruction depends on appropriate selection of the components of arithmetic to be used in assessment and intervention. This is still an issue for debate and one which requires considerable further research. One of the main potential problems, which was more common in the past than nowadays, is to assume that the components to be addressed must necessarily correspond to specific arithmetical operations: e.g. treating "addition", "subtraction", "multiplication", "division" etc. as separate components. It is, of course quite possible for children to have specific problems with a particular arithmetical operation. Indeed, as we have seen, it is possible for a particular arithmetical operation to be selectively impaired in adult patients following brain damage. Nevertheless, it is an over-simplification to assume that these operations are likely to be the primary components of arithmetical processing. Current classifications tend to place greater emphasis on the type of cognitive process; e.g. the broad distinctions between factual knowledge ("knowing that"), procedural knowledge ("knowing how"), conceptual knowledge ("knowing what it all means") and in some theories utilizational knowledge ("knowing when to apply it") (see, for example, Greeno, Riley and Gelman, 1984). A potential danger of over-emphasizing the different operations as separate components is that it may encourage children, and perhaps adults, to ignore the relationships between the different operations. Another potential problem - again commoner in the past though still a danger nowadays - is looking at children's difficulties only in terms of procedural errors. It is, of course, important to investigate the strategies that individual children use in arithmetic, including those faulty arithmetical procedures that lead to errors. Nonetheless, diagnosing the incorrect strategies is not always the final step. There may be a conceptual reason why the incorrect strategy is acquired and maintained or there may be unperceived conceptual strengths, which need to be noted and built on (Tilton, 1947; Ginsburg, 1977).
Such diagnostic work is vital. Children do indeed frequently acquire incorrect strategies, which can become entrenched, especially if the child is given too much of the wrong sort of arithmetical practice. Nonetheless, diagnosing the incorrect strategies is not always the final step. There may be a conceptual reason why the incorrect strategy is acquired and maintained. In the case of the faulty subtraction strategies described above, their acquisition could result from an assumption that larger numbers cannot be subtracted from smaller numbers, combined with an inadequate understanding of place value which makes it difficult for children to understand the nature and purpose of borrowing. As Tilton (1947, p. 85) goes on to remark, "Many of the errors made by these... children seemed to be due to an insufficient understanding of the meaning of numbers. It seems as if these children had been asked to learn the rules for the manipulation of numbers in addition, subtraction and multiplication without having learned the meaning of the symbols that they have been asked to manipulate". More generally, interventions need to take into account the fact that arithmetical ability is made up of many components. Ginsburg (1972) pointed out that "children's knowledge of mathematics is extraordinarily complex and often much different from what we had supposed it to be... In the case of every child we have interviewed or observed, there have emerged startling contradictions, unsuspected strengths or weaknesses, and fascinating complexities". Of course, no intervention programme can take into account all possible components, but they are likely to be most effective if not restricted to one or two components, and if they allow for different children having different types of difficulty, which may not be restricted to procedural difficulties.
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