Which aspects of arithmetic are most likely to create problems?

Despite the variable patterns discussed above, here are particular areas of arithmetic that do appear to create more problems than others for children. One of the areas most commonly found to create difficulties is memory for arithmetical facts. Studies of children with mathematical difficulties show them to be more consistently weak at retrieving arithmetical facts from memory than at other aspects at arithmetic. They often rely on counting strategies in arithmetic at ages when their age-mates are relying much more on fact retrieval (Russell and Ginsburg, 1984; Siegler, 1988; Geary and Brown, 1991; Ostad, 1997, 1998; Cumming and Elkins, 1999; Fei, 2000).

Jordan, Hanich and Kaplan (2003) tested American second grade children for knowledge of addition and multiplication facts. 45 children with poor arithmetic fact mastery were compared with 60 children with good arithmetic fact mastery. They were followed up longitudinally through second and third grade. The children with poor fact mastery showed little improvement on timed number fact tests in over a year, but showed normal progress in other aspects of mathematics. When IQ was held constant, the children with poor fact mastery performed similarly with good fact mastery in tests of reading and mathematics word problem solving at the end of third grade.

Thus, difficulties in memory for arithmetic facts tend to be persistent. They appear to be independent of reading skills, and did not affect performance on other aspects of arithmetic. Part of the reason for the associations between number fact retrieval and more general arithmetical performance lies in the ways in which arithmetical performance is often assessed. If arithmetical tests and assessments emphasize fact retrieval, then those who are poor at fact retrieval are likely to do badly in the tests, and be classed as having arithmetical difficulties. If arithmetical fact retrieval is emphasized in the school curriculum, then those who are weak at this aspect of arithmetic will struggle with their school arithmetic lessons and assignments, even if they have no difficulty with other aspects of arithmetic.

It does, however, seem that arithmetic fact retrieval difficulties have effects beyond tasks that emphasize such facts, and spill over into other areas of mathematics. If people have trouble in remembering basic arithmetic facts, then they will have to calculate these facts by alternative and usually more time-consuming strategies. Even if they are able to do so accurately, this means that they must devote time and attention to obtaining facts that someone else might retrieve automatically; and this will divert time and attention from other aspects of arithmetical problem-solving, resulting in lower efficiency.

There do seem to be different degrees and forms of difficulty with number facts and resulting reliance on counting strategies. Some children, such as those in the studies by Russell and Ginsburg (1984) and Jordan et al (2003), seem to have specific, localized difficulties with fact retrieval, and to be able to use a wide variety of alternative strategies. Other children (Gray 1997; Ostad, 1997, 1998) seem to rely narrowly and exclusively on counting strategies, and fail to use any other form of strategy, including derived fact strategies. To add to their difficulties, such children are often less efficient than others at using the very counting strategies on which they rely the most.

For example, Ostad (1997) studied Norwegian children with mathematical difficulties. The study included 32 children with and 32 children without mathematical difficulties in Grade 1; 33 children with and 33 children without mathematical difficulties in Grade 3; and 36 children with and 36 children without mathematical difficulties in Grade 5. The children with mathematical difficulties were those who scored below the 14th percentile on a Norwegian standardized mathematics achievement test. The pupils were asked to solve 28 single-digit addition problems on two different occasions separated by a period of two years. Their strategies on each problem were recorded. The children with mathematical difficulties used almost exclusively counting based strategies, while those without such difficulties children were more likely to use retrieval or derived fact strategies. Moreover, children without mathematical difficulties increased their use of retrieval and decreased their use of counting-based strategies as they grew older, while the strategies of the children with mathematical difficulties did not change with age. At all ages, children without mathematical difficulties used a far wider variety of strategies than those with mathematical difficulties, and the differences increased with age.

It should, however, be noted that, while difficulty in remembering number facts is a very common component of arithmetical difficulties, not all children with arithmetical difficulties have this problem. Most of the children in Dowker’s intervention study (Dowker, 2004, in press) certainly did experience problems with number facts; but not all did. Some children could remember many number facts, but seemed to lack strategies (including suitable counting strategies) for working out sums when they did not know the answer. Some other children could deal with single-digit arithmetic but had serious difficulty in achieving even limited understanding of tens, units and place value. Russell and Ginsburg (1984) found that difficulties with word problem solving, as well as with memory for facts, characterized 9-year-old children who were described by their teachers as weak at arithmetic. In this study, most were not significantly worse than others at tasks involving estimation, derived fact strategy use, or understanding the relationships between tens and units.

Bryant, Bryant and Hammill (2000) found that several difficulties were common in children with mathematical weaknesses, but that the commonest problem was a difficulty in carrying out multi-step arithmetic. They carried out a large-scale study of the characteristics of children and young people with mathematical difficulties. The participants were 1724 American pupils from 8;0 to 18;11, diagnosed as learning disabled and receiving special education services. 870 were rated by their teachers as having mathematical weaknesses; 854 were not. The researchers constructed a list of 33 mathematical behaviours derived from consulting the literature on developmental and acquired dyscalculia. Items included "difficulty with word problems"; "difficulty with multi-step problems"; "does not recognize operator signs"; "does not verify numbers, and settles for first answer"; etc. Teachers were asked to check the items on the list that applied to each pupil. Stepwise multiple regression showed that just under 31% of the variance between the groups with and without mathematical weaknesses was caused by a single item: "Has difficulty with multi-step problems and makes borrowing errors". 7 other items contributed significantly to group membership (though to a much lesser extent, only accounting in total to just under 5% of the variance). These were: "Cannot recall number facts automatically"; "Misspells number words; "Reaches unreasonable answers"; "Calculates poorly when order of digits is altered"; "Cannot copy numbers accurately"; "Orders and spaces numbers inaccurately in multiplication and division"; and "Doesn't remember number words."

The discussion so far has concerned arithmetical processes, concepts and skills that are important from the earliest stages of arithmetical learning. Weaknesses in these areas are noticeable in primary school children, though they often persist into adolescence and adulthood. Weaknesses can, and frequently do, arise in various components of secondary school mathematics. Hart (1981) and her team found that secondary school pupils have many difficulties, both procedural and conceptual, with many mathematical topics, including ratio and proportion; fractions and decimals; algebra; and problems involving area and volume. She

concluded (p. 209) that “mathematics is a very difficult subject for most [secondary school] children” and that “understanding improves only slightly as the child gets older”. Since difficulties that are specific to secondary school mathematical topics are indeed very frequent, they are not usually seen as special difficulties in mathematics, and are not the subject of this review. It must be remembered, however, that if early difficulties are not remediated, they are likely to result in very severe difficulties with those more advanced topics that tend to present problems for many people.