Arithmetical difficulties in relation to verbal and spatial ability

Reasoning, including arithmetical reasoning, can be carried out in many ways. Two broad categories that are often discussed with regard to individual differences are verbal and spatial reasoning. Information can be represented, manipulated and analysed in words; it can also be represented, manipulated and analysed in terms of visual-spatial imagery. A number of researchers have investigated the issue of whether arithmetical skills are particularly associated with verbal or spatial reasoning and/or with discrepancies between the two. The factor analytic studies used to construct some commonly used IQ scales have consistently placed the Arithmetic subtest (one which emphasizes word problem solving) within the Verbal scale. However it has sometimes been suggested that spatial difficulties are particularly associated with difficulties in arithmetical reasoning.

 Rourke (1983, 1993) proposed that verbal weaknesses lead to memory difficulties and that nonverbal weaknesses lead to logical difficulties. He proposed two basically different groups of children with arithmetical learning difficulties. Children in the first group have difficulties in retrieval of number facts and in working memory, but have a reasonably good understanding of number concepts. They have higher nonverbal than verbal IQs and often have difficulties with reading as well as mathematics. Children in the second group do not have memory problems but do have conceptual problems; they have higher verbal than nonverbal IQs; are less likely to have reading or language difficulties, but more likely to have spatial and social difficulties associated with right hemisphere deficits. A few studies by Rourke and others (e.g. Robinson, Menchetti and Torgesen, 2002) have supported the view that children with both reading and mathematical deficits tend to have more memory difficulties but fewer conceptual difficulties than those with just mathematical deficits. However, there has been no consistent support for the view that 'left hemisphere'-type verbal deficits are associated with procedural and factual memory difficulties in arithmetic, while 'right-hemisphere'-type nonverbal deficits are associated with conceptual difficulties inarithmetic. Shalev, Manor, Amir, Weirtman and Gross-Tsur (1997) found no differences in the types of mathematical difficulty demonstrated by dyscalculic children with higher verbal versus higher non-verbal IQ.

Jordan and Hanich (2000) studied 76 American second-grade children. They were divided into four achievement groups: 20 children with normal achievement in reading and mathematics; 10 children with difficulties in both reading and mathematics (MD-RD), 36 children with difficulties in reading only (RD) and 10 children with difficulties in mathematics only (MD). They were given tests of four areas of mathematical thinking: number facts, story problems, place value and written calculation. Children with MD/RD performed worse than NA children on all aspects of mathematics; those with MD performed worse than NA children only on story problems.

Hanich, Jordan, Kaplan and Dick (2001) similarly divided 210 second-graders into four achievement groups: children with normal achievement in reading and mathematics; children with difficulties in both reading and mathematics (MD-RD), children with difficulties in reading only (RD) and those with difficulties in mathematics only (MD). Both MD groups performed worse than the other groups in most areas of arithmetic. The MD-only group outperformed the MD-RD group in both exact mental calculation and problem solving. The two MD groups performed similarly on written calculation, place value understanding, and approximate arithmetic.

Geary, Hoard and Hamson (1999) studied 90 first-grade children in the average IQ range. They included 35 children with normal achievement in reading and mathematics (N); 15 children with mathematical difficulties (MD; as shown by scores below the 30th percentile on the Mathematical Reasoning subtest of the Wechsler Individual Achievement Test); 15 children with reading difficulties (RD; as shown by scores below the 30th percentile on the Word Attack subtest of the Woodcock Johnson Psycho-Educational Battery; and 25 children with both mathematical and reading difficulties (MD/RD). Both MD groups showed problems in fact

retrieval and in using counting strategies correctly in arithmetic. Children who had difficulties with both mathematics and reading tended to show problems in understanding counting principles and detecting counting errors; those with only MD or RD did not. However, about half of the MD children made double-counting errors. The MD/RD children, and those MD children who made double-counting errors, had lower backward digit spans than the other children.

So, the studies by Jordan and her colleagues and by Geary et al (1999) suggest that children with combined mathematical and reading disabilities tend to perform badly on more aspects of mathematics than children who only have mathematical difficulties; but do not support the type

of dichotomy suggested by Rourke. 

Thus, while signs of verbal or spatial weaknesses should serve as a warning signal that a child may experience mathematical difficulties, they cannot be used as definite predictors of either the existence or type of mathematical difficulty that a child may have. There is still less evidence that, within the general population, verbal and nonverbal ability are associated with consistently different forms of strengths and weaknesses within arithmetic. (This is not to say that there might not be such patterns within the broader domain of mathematics; e.g. geometry is likely to be more specifically associated with spatial ability than is arithmetic). Dowker (1995, 1998) looked at WISC IQ scores, calculation and derived fact strategy use in 213 children between the ages of 6 and 9. Both Verbal and Performance I.Q. predicted performance on tasks of both arithmetical calculation and derived fact strategy use. Verbal I.Q. was a stronger predictor than Performance I.Q. of both types of arithmetical task. Children who showed a strong discrepancy between verbal and nonverbal I.Q. in either direction tended to do well at tasks that involve the use of derived fact strategies; such discrepancies did not predict calculation performance.