Ann Dowker

Early Identification and Intervention for Students With Mathematics Difficulties

The extremely interesting article of Gersten, Jordan, and Flojo (in this issue) discusses the early identification of mathematics difficulties (MD) in children; the importance of early intervention; relationships between mathematical and reading difficulties; and some important early indicators of MD in young children. Perhaps the most important conclusion of the article is that it is important to identify early signs and predictors of MD to ameliorate and perhaps prevent later MD. This commentary will discuss other research in the area, which for the most part supports the arguments and proposals put forward by Gersten et al. In fact, the body of research on MD relevant to early identification and early intervention is not quite as small as is often thought. It is true that far less research has been carried out on mathematical difficulties than on reading difficulties. However, far more has been carried out than is sometimes recognized, possibly because the research has been carried out over a long period, in different countries, and within different disciplines (education, psychology, and neuroscience). People are often unaware of relevant research that has been carried out in other countries, other disciplines, or at an earlier time. This situation has unfortunately limited the scope and application of such research. It should be noted that although the article, this commentary, and much of the research in the area speak of mathematical difficulties, it would be more strictly appropriate to speak of arithmetical difficulties. Mathematics includes many other topics, such as geometry, measurement, and algebra. However, most studies of young children’s MD have dealt mainly with number and arithmetic. This commentary will discuss (a) research on the nature and heterogeneity of MD; (b) research on early screening and intervention; (c) some ongoing research on early predictors of specific mathematical strengths and weaknesses.

Gender differences in mathematics anxiety and the relation to mathematics performance while controlling for test anxiety.

BackgroundMathematics anxiety (MA), a state of discomfort associated with performing mathematical tasks, is thought to affect a notable proportion of the school age population. Some research has indicated that MA negatively affects mathematics performance and that girls may report higher levels of MA than boys. On the other hand some research has indicated that boys’ mathematics performance is more negatively affected by MA than girls’ performance is. The aim of the current study was to measure girls’ and boys’ mathematics performance as well as their levels of MA while controlling for test anxiety (TA) a construct related to MA but which is typically not controlled for in MA studies.MethodsFour-hundred and thirty three British secondary school children in school years 7, 8 and 10 completed customised mental mathematics tests and MA and TA questionnaires.ResultsNo gender differences emerged for mathematics performance but levels of MA and TA were higher for girls than for boys. Girls and boys showed a positive correlation between MA and TA and a negative correlation between MA and mathematics performance. TA was also negatively correlated with mathematics performance, but this relationship was stronger for girls than for boys. When controlling for TA, the negative correlation between MA and performance remained for girls only. Regression analyses revealed that MA was a significant predictor of performance for girls but not for boys.ConclusionsOur study has revealed that secondary school children experience MA. Importantly, we controlled for TA which is typically not controlled for in MA studies. Girls showed higher levels of MA than boys and high levels of MA were related to poorer levels of mathematics performance. As well as potentially having a detrimental effect on ‘online’ mathematics performance, past research has shown that high levels of MA can have negative consequences for later mathematics education. Therefore MA warrants attention in the mathematics classroom, particularly because there is evidence that MA develops during the primary school years. Furthermore, our study showed no gender difference in mathematics performance, despite girls reporting higher levels of MA. These results might suggest that girls may have had the potential to perform better than boys in mathematics however their performance may have been attenuated by their higher levels of MA. Longitudinal research is needed to investigate the development of MA and its effect on mathematics performance.

Computational estimation strategies of professional mathematicians

Forty-four academic pure mathematicians were given Levines (1982) computational estimation task, which involved estimating the answers to 20 multiplication and division problems and describing the strategies used. The mathematicians were accurate estimators, and they used a great variety of strategies, as many as 23 for a single problem. All but 2 of the mathematicians used some strategy that was not used by the other 43 for at least one problem. When 18 mathematicians were retested after an interval of several months, they used different strategies on the second test for 9 to 17 of the 20 problems. Some theoretical implications of these findings are discussed.

Estimation Strategies of Four Groups.

Selected groups of 44 academic “pure” mathematicians, 44 accountants, 44 psychology students, and 44 English students were given Levines (1982) computational estimation task, which involved mentally estimating the products and quotients of 20 multiplication and division problems and describing their strategies. The mathematicians were the most accurate estimators, and the English students the least accurate, with psychology students and accountants obtaining similar scores intermediate between the other groups. All groups demonstrated an impressively versatile use of appropriate strategies. The mathematicians and accountants used significantly larger numbers of appropriate strategies than the other groups and strongly resembled one another in this respect, despite the significantly greater accuracy of the mathematicians. All the non-mathematician groups used significantly larger numbers of inappropriate strategies than did the mathematicians. We discuss (1) the implications for cognitive psychology of th...

Cognitive Enhancement or Cognitive Cost: Trait-Specific Outcomes of Brain Stimulation in the Case of Mathematics Anxiety

The surge in noninvasive brain stimulation studies investigating cognitive enhancement has neglected the effect of interindividual differences, such as traits, on stimulation outcomes. Using the case of mathematics anxiety in a sample of healthy human participants in a placebo-controlled, double-blind, crossover experiment, we show that identical transcranial direct current stimulation (tDCS) exerts opposite behavioral and physiological effects depending on individual trait levels. Mathematics anxiety is the negative emotional response elicited by numerical tasks, impairing mathematical achievement. tDCS was applied to the dorsolateral prefrontal cortex, a frequent target for modulating emotional regulation. It improved reaction times on simple arithmetic decisions and decreased cortisol concentrations (a biomarker of stress) in high mathematics anxiety individuals. In contrast, tDCS impaired reaction times for low mathematics anxiety individuals and prevented a decrease in cortisol concentration compared with sham stimulation. Both groups showed a tDCS-induced side effect—impaired executive control in a flanker task—a cognitive function subserved by the stimulated region. These behavioral and physiological double dissociations have implications for brain stimulation research by highlighting the role of individual traits in experimental findings. Brain stimulation clearly does not produce uniform benefits, even applied in the same configuration during the same tasks, but may interact with traits to produce markedly opposed outcomes.

Young children's Addition Estimates

Two hundred and fifteen children aged between 5 and 9 years were asked to estimate the answers to addition sums. Their competence at addition was first assessed, and they were accordingly divided into five groups. Children of each level were given a set of estimation problems involving sums a little too difficult for them to calculate (termed their “base correspondence”). Of the 215 children, 108 were then given sets of estimation problems corresponding to levels higher than their own. In the base correspondence, children of higher levels tended to produce more reasonable estimates than did children of lower levels. As difficulty increased beyond the base correspondence, the reasonableness of the estimates declined. The existence and nature of a zone of partial knowledge and understanding are discussed.

Dyscalculia from a developmental and differential perspective.

Developmental dyscalculia (DD) and its treatment are receiving increasing research attention. A PsychInfo search for peer-reviewed articles with dyscalculia as a title word reveals 31 papers published from 1991–2001, versus 74 papers published from 2002–2012. Still, these small counts reflect the paucity of research on DD compared to dyslexia, despite the prevalence of mathematical difficulties. In the UK, 22% of adults have mathematical difficulties sufficient to impose severe practical and occupational restrictions (Bynner and Parsons, 1997; National Center for Education Statistics, 2011). It is unlikely that all of these individuals with mathematical difficulties have DD, but criteria for defining and diagnosing dyscalculia remain ambiguous (Mazzocco and Myers, 2003). What is treated as DD in one study may be conceptualized as another form of mathematical impairment in another study. Furthermore, DD is frequently—but, we believe, mistakenly- considered a largely homogeneous disorder. Here we advocate a differential and developmental perspective on DD focused on identifying behavioral, cognitive, and neural sources of individual differences that contribute to our understanding of what DD is and what it is not.

Numeracy recovery: a pilot scheme for early intervention with young children with numeracy difficulties

The Numeracy Recovery scheme described here involves working with children who have been identified by their teachers as having problems with arithmetic. These children are assessed on eight components of early numeracy: (1) principles and procedures related to counting; (2) use of written arithmetical symbolism; (3) use of place value in arithmetic; (4) understanding and solution of word problems; (5) translation between concrete, verbal and numerical formats; (6) use of derived fact strategies for calculation; (7) arithmetical estimation and (8) memory for number facts. The children received weekly individual intervention in the particular components with which they were found to have difficulty. For the first 62 children in the project, Wilcoxon tests showed a significant improvement in standardised scores in both tests following intervention.

Interventions for improving numerical abilities: Present and future

Abstract Low numeracy skills have a negative impact on the employment prospects and mental and physical health of individuals, and on the economic status of countries. Clearly, this is a high priority area where efficient strategies for intervention can lead to a better outcome, especially when implemented at an early age. We discuss here present and future directions for intervention. The development of such interventions has been based on the study of numerical difficulties through methods ranging from standardized tests to behavioral measures to neuroimaging. The intervention techniques range from group-based interventions targeted at strengths and weaknesses in specific components of arithmetic, to educational computer-games, to non-invasive brain-stimulation. We discuss the principles behind each method, the current evidence, and future directions.

Linguistic Influences on Mathematical Development: How Important Is the Transparency of the Counting System?

Wales uses languages with both regular (Welsh) and irregular (English) counting systems. Three groups of 6- and 8-year-old Welsh children with varying degrees of exposure to the Welsh language—those who spoke Welsh at both home and school; those who spoke Welsh only at home; and those who spoke only English—were given standardized tests of arithmetic and a test of understanding representations of two-digit numbers. Groups did not differ on the arithmetic tests, but both groups of Welsh speakers read and compared 2-digit numbers more accurately than monolingual English children. A similar study was carried out with Tamil/English bilingual children in England. The Tamil counting system is more transparent than English but less so than Welsh or Chinese. Tamil-speaking children performed better than monolingual English-speaking children on one of the standardized arithmetic tests but did not differ in their comparison of two-digit numbers. Reasons for the findings are discussed.