Individual Differences in Arithmetic

Implications for Psychology, Neuroscience and Education

Ann Dowker

• Publisher: Routledge
• Year: 2019
• ISBN:
  9781317627425
• Edition: 2
• Topic: Psicología
• Subtopic: Psicología cognitiva y cognición

Arithmetic is still hugely important in many aspects of modern life, but our personal attitudes to it differ greatly. Many people struggle with the basic principles of arithmetic, whilst others love it and feel confident in their arithmetical abilities. Why are there so many individual differences in people's performance in, and feelings about, arithmetic?

Individual Differences in Arithmetic explores the idea that there is no such thing as arithmetical ability, only arithmetical abilities. The book discusses several important components of arithmetic, from counting principles and procedures to arithmetical estimation, alongside emotional and cognitive components of arithmetical performance. This edition has been extensively revised to include the latest research, including recent cross-cultural and cross-linguistic research, the development of new interventions for children with difficulties and studies of early foundations of mathematical abilities.

Drawing on developmental, educational, cognitive and neuropsychological studies, this book will be essential reading for all researchers of mathematical cognition. It will also be of interest to educators and other professionals working within individuals with arithmetic deficits.

1

INDIVIDUAL DIFFERENCES IN ARITHMETIC

Children, adults, males, females, weaknesses and talents

Most people can do maths when they’re 4, if they start school when they’re 4. And when they’re grown up, some people can do maths and some can’t.

(Tom, aged 6)

It is well known that individual differences in arithmetical performance are very marked in both children and adults. Kaufmann and Nuerk (2005) claimed that ‘… average arithmetic development does not pursue a straight, fully predictable course of acquisition, but rather can be characterized by quite impressive individual differences’.

For example, Cockcroft (1982) reported that an average British class of 11-year-olds is likely to contain the equivalent of a 7-year range in arithmetical ability. Despite many changes in UK education since then, including the introduction of a standard National Curriculum and a National Numeracy Strategy, almost identical results were obtained by Brown et al. (2002). They found that the gap between the 5th and 95th percentiles on standardized mathematics tests by children in Year 6 (10- to 11-year-olds) corresponded to a gap of about 7 years in ‘mathematic ages’. Individual differences in arithmetic among children of the same age are also very great in most other countries, though some studies suggest that they are less pronounced in Pacific Rim countries (Mullis, Martin, Goh, & Cotter, 2015).

Individual differences in arithmetic are easier to detect in children, who are attending school and taking mathematics tests, but it is clear that they persist throughout life. Some adults have severe difficulties with basic numeracy, and most such adults were already struggling with arithmetic by the age of 7 (Bynner & Parsons, 1997; Parsons & Bynner, 2005). By contrast, others have a fascination with numbers, are exceptionally skilled calculators and/or reason exceptionally well about numbers. A few are ‘lightning calculators’ who can perform multi-digit calculations in their heads as fast as or faster than the average calculating machine.

Environment, heredity and arithmetical ability

There are numerous environmental influences on arithmetical development. Culture, education and even the language that we speak all play an important part in the way that arithmetic is acquired. These influences will be discussed in Chapter 9 of this book. More rarely, illness or injury can cause brain damage, which has a significant effect on arithmetic. Perinatal factors, such as severe prematurity, can have gross or subtle effects on brain development which influence arithmetic (Isaacs, Edmonds, Lucas, & Gadian, 2001; Simms et al., 2015; Tasaka & Shimada, 2000). The effects of brain damage and unusual patterns of brain development will be discussed in Chapter 10.

As regards genes, there are a few genetic disorders, such as Turner’s syndrome and Williams syndrome, which are particularly associated with arithmetical difficulties. Such disorders are rare, but there is a tendency for arithmetical difficulties to run in families, even when not associated with a known genetic disorder. For example, Knopnik, Alarcon and DeFries (1997) found that monozygotic twins were significantly more likely than dizygotic twins to be concordant for mathematical disabilities. There was also a significant genetic correlation between mathematical disabilities and reading disabilities.

The possible genetic contribution to arithmetical disabilities will be discussed further in Chapter 10. There has been a relatively small, but increasing, amount of research into the possible genetic contribution to variation in arithmetical performance in the population as a whole.

Heritability is defined as the proportion of variance in a particular characteristic for a given population that can be explained by heredity. It is usually investigated by comparing levels of similarity in identical versus non-identical twins and/or in biological versus adopted relatives. There is considerable controversy both about the accuracy and appropriateness of the methods used to estimate heritability, and about the validity of even attempting to separate genetic from environmental contributions to individual differences, since there are undoubtedly strong interactions between genetic and environmental factors.

Bearing these limitations in mind, the relevant studies suggest that there is a significant genetic contribution to individual differences in mathematics. Vandenberg (1966) carried out a twin study and obtained a heritability estimate for mathematical ability of approximately 50%. More recent twin studies have also suggested significant heritability for mathematical ability (Davis, Haworth, & Plomin, 2009; Hart, Petrill, & Kamp Dush, 2010; Kovas et al., 2013; Oliver et al., 2004). Heritability estimates in the various studies range from 0.32 to 0.72, seemingly dependent on the tasks used and on the country where the research is being done. On the whole, British studies tend to give higher heritability estimates than American studies. This may possibly be because Britain has a National Curriculum, and thus the environment is more uniform with regard to mathematical experiences, meaning that a lower proportion of variance is due to environment (Tosto, Haworth, & Kovas, 2015).

Further evidence for some genetic component to mathematical difficulties comes from a study of siblings of children with mathematical disabilities. Desoete, Praet, Titeca and Ceulemans (2013) compared 9 siblings of children with diagnosed mathematical learning difficulties with 63 age-matched children with no family members with known mathematical learning difficulties. They were all given three number line estimation tests involving sets of (1) dots, (2) numerals and (3) number words. The siblings of the children with mathematical difficulties performed significantly worse than the other children on all these tests, especially those involving numerals and number words. Although one cannot rule out some environmental influences affecting children in the same family, the study does suggest that those who are genetically related to those with mathematical learning difficulties are likely to have relative weaknesses in numerical abilities, even if they do not always amount to full-blown mathematical disabilities.

The major debates about genetic and environmental influences on mathematical ability have tended to focus not so much on population variance as on gender differences. Although gender differences are not in fact great in most aspects of mathematics, they have been an intense focus for research in the field of individual differences in mathematics. This topic must therefore be considered at some length.

Gender differences in arithmetic

“Girls [or boys] go to Mars

To get more stars.

Boys [or girls] go to Jupiter

To get more stupider.”

(British playground rhyme, reversible according to the gender of the speaker.)

I do wish I could do clock sums and those sums about taps running in and out. I wonder whoever invented such nonsense. I wish I could give the man who thought of them my honest opinion of his wits.

How do you know it was a man?’ inquired my brother … ‘It is girls who think of ridiculous things.

Not sums, especially pipe sums. What girl would let water in by two taps and out by one if she wanted to fill a bath? It takes a boy to think of that way of doing things.

(Maude Forsey: Mollie Hazeldene’s Schooldays; Nelson, 1924, p. 16)

Are males better at mathematics than females?

The folk wisdom is that males are better at mathematics than females. There is little evidence for this in the population as a whole. Groups selected as being to be extremely good at mathematics do tend to be predominantly male (Benbow, 1988). Certainly, significantly more males than females are to be found in advanced undergraduate mathematics courses, and the large majority of professional research mathematicians are men. However, Spelke (2005) notes a significant increase over time in the number of American female undergraduates studying mathematics. At the other end of the scale, most studies indicate that serious mathematical difficulties are equally common in males and females (Gross-Tsur, Manor, & Shalev, 1996; Lewis, Hitch, & Walker, 1994), and Reigosa-Crespo et al. (2012) found that the combination of severe mathematical difficulties with problems in basic number representation was actually commoner in boys (see section in this chapter on ‘Dyscalculia: what is it and is it a separate condition?’).

Gender differences in mathematics have decreased overall over the year, as shown by the successive TIMSS (Trends in International Mathematics and Science Study) international studies of pupils’ mathematical performance (Mullis et al., 2015; Mullis, Martin, Foy, & Hooper, 2016a; Mullis, Martin, & Loveless, 2016b). When international surveys combine data from many countries, there is still some advantage in secondary school in favour of boys, but this is small and is not found in all countries, and a few countries such as Jordan and Malaysia show an advantage for girls.

Spatial ability and gender

Spatial ability has frequently been suggested (e.g. by Casey, Nuttall, & Benbow, 1995) to be an important factor in mathematical performance in general, and in gender differences in mathematics in particular.

There is no doubt that gender differences in spatial ability are frequently observed, though they tend to be quite small and are mainly found for tasks involving three-dimensional than two-dimensional space (Voyer, Voyer, & Bryden, 1995). There has been much debate as to whether such gender differences are due to biological factors (Geary, 1996) or to environmental factors such as boys being given more constructional toys and more freedom to explore their environment independently. Can this explain gender differences in mathematics?

Relationships between spatial and arithmetical abilities will be discussed later in this chapter. Though there is much evidence for spatial representation being involved in arithmetic, it is still controversial whether there are strong links between three-dimensional spatial abilities and non-geometrical mathematical abilities.

It is sometimes suggested that there may actually be gender differences with regard to the extent to which spatial ability predicts mathematical performance, but findings are contradictory with regard to the nature of such gender differences. Some studies (Casey, Pezaris, & Nuttall, 1992; Connor & Serbin, 1985) have found that spatial ability predicts mathematical performance more in males than in females, while the opposite was found by Friedman (1995).

Thus, if arithmetic is indeed influenced by gender differences in spatial ability, the influence is likely to be quite limited and subtle.

Attention, mathematics and gender

As suggested by Dowker (1996), future consideration of gender differences in mathematics should address the issue of single-minded concentration. Personal communications from several male and female mathematicians suggest that one important factor in mathematical research, to a greater extent than for other academic subjects, is the ability to concentrate single-mindedly on a given problem.

There is some evidence that women may be more able than men to pay attention to several topics at once, and less willing or able to focus attention in the exclusive way demanded, for example, by advanced mathematics. There are obvious social reasons why this may be the case: women in their traditional role as caregiver to children may need to be able to divide their attention between their own tasks and several children’s different needs and requests. There may also be biological factors involved. Males are more likely than females to be diagnosed as having ‘attention deficit disorder’. Though this condition is probably overdiagnosed in some places, and there are uncertainties as to the criteria for diagnosis, this difference does suggest that males may be more likely than females to be found at either extreme of the ability to focus attention narrowly.

Mathematics self-concept, mathematics anxiety and gender

As discussed in Chapter 11, mathematics arouses anxiety and other negative emotions in many people: often linked to low estimates of their own ability and expectations of failure. There is also evidence that females tend to experience more anxiety than males with respect to mathematics, and to rate themselves lower in mathematics than do males. Studies tend to show that boys like mathematics more than girls do, and that girls’ attitude to mathematics declines more over time than that of boys. However, the gender difference in attitude may be less now than some years ago.

Our own research has found no overall gender differences in primary school children’s liking for mathematics (Dowker, Bennett, & Smith, 2012; Chapter 11 of this book). However, girls do tend to give themselves lower ratings in mathematics than boys do.

It is often suggested that females’ greater anxiety and lower self-concept in mathematics may lead to poorer performance and less inclination to choose to study mathematics.

The question is which comes first. Do females find mathematics more difficult for other reasons, and therefore experience more anxiety about it; or is the difficulty due to the anxiety? If the anxiety comes first, then why are females more anxious? They may be more liable to performance anxiety in general, but then why is their literacy development not equally affected by anxiety? One possible explanation is that they experience more negative reactions to their mathematical performance from adults and peers as a result of social stereotypes. If a boy performs badly at a mathematics test, this may be attributed to a temporary cause: he was not feeling well on the day, or he did not understand particular questions, or he had not revised sufficiently for the test. If a girl performs badly at the same test, she may be more quickly labelled as ‘no good at maths’.

Of course, boys usually can’t understand algebra!

(Remark made by a female teacher to the parents of a 14-year-old boy, Britain, 2002)

In Britain, some of the gender-related expectations about mathematical performance may have diminished or even reversed in recent years. As boys’ general academic performance has declined in comparison with girls, teachers and policymakers have become increasingly concerned with male underachievement. Until the 1980s, books and papers on gender issues in education tended to be concerned mainly with the disadvantages experienced by girls. Since the 1990s, there have been far more writings on the educational difficulties and disadvantages experienced by boys. It would be of interest to investigate whether this has changed adults’ perception of gender differences in mathematics. Anecdotal evidence – e.g. the remark that heads this paragraph – suggests that this is a possibility, though gender stereotypes about females being worse at mathematics are still prevalent (see Chapter 11 for more discussion of such stereotypes and their influences on mathematics anxiety).

People with difficulties in arithmetic

There will be many references in this book to children and adults with ‘difficulties’ in arithmetic, mathematics or numeracy. The term simply refers to children or adults who struggle or fail to cope with some of the aspects of arithmetic which are necessary or desirable for educational or practical purposes. Since the term ‘learning difficulties’ is used in some educational and clinical contexts to imply severe and unusual problems that may require special education, it should be emphasized that the use of the term ‘difficulties’ does not have such an implication here. It does not mean that these difficulties are pathological, that they are always classifiable as ‘specific learning difficulties’ in the technical sense or that they must be either innate or the result of brain damage. Some difficulties with arithmetic do indeed involve a pathological process, as in the case of adults who lose previously established arithmetical concepts or skills as the result of brain damage. However, most difficulties in arithmetic, like most difficulties in learned subjects, lie on a ‘normal’ continuum between extreme talent and extreme weakness, and are due not to brain damage but to a mismatch between an individual’s pattern of cognitive strengths and weaknesses and the way that (s)he is taught.

Dyscalculia: what is it, and is it a separate condition?

In recent years, there has been increased emphasis on the possibility that, just as some children have dyslexia, some may have dyscalculia: a specific difficulty in doing arithmetic (Butterworth, Varma, & Laurillard, 2011; Kuhn, 2015; Landerl, 2013).

There are a number of studies that investigate the nature and prevalence of specific learning difficulties in mathematics. For example, Lewis et al. (1994) studied 1,056 unselected 9- to 10-year-old English children (the entire age group within a particular, socially highly heterogeneous, local education authority, excluding only those assessed as having severe general learning difficulties). They were given Raven’s Matrices IQ Test, Young’s Group Mathematics Test and Young’s Spelling and Reading Test. 1.3% of the sample had specific arithmetical difficulties, defined as an arithmetic scaled score of 85 or below, despite a Raven’s IQ score of 90 or above. A further 2.3% had difficulties in both reading and arithmetic (scaled scores of 85 or below in both the reading and arithmetic tests) despite a Raven’s IQ score of 90 or above. Thus, the prevalence of arithmetic difficulties in children of at least average cognitive ability was 3.6%. The children with arithmetical difficulties were equally divided as to gender, which contrasts with the general finding that boys are far more likely than girls to have language and literacy difficulties.

Gross-Tsur, Manor and Shalev (1996) assessed the incidence of dyscalculia in a cohort of 3,029 Israeli 11- to 12-year-olds. The 600 children who scored in the lowest 20% on a standardized c...