New Voices: 'It’s like plaiting fog'

I wasn't a great mathematician at school but I managed. I got a 'B'. When I became an undergraduate in psychology, again I managed. Day to day I know when I am late, I know when I have a limited time on a task, I can count my daily calorie intake (when I can be bothered) and I can lighten my purse by counting out all the change just so I don't have to break into a 10 pound note. I also know approximately how much is in my bank account at any given time. In essence, I take seeing and understanding number for granted. And I use the skill every day.

It was puzzling for me then, 12 years ago as an undergraduate, when I found myself sitting with a 15-year-old student in my role as a classroom assistant trying to explain how to divide a number by three. 'Katy' had one year left of school, and simply didn't 'get maths'. When I talked to her teacher about the problem he agreed, stating simply 'I know… It's like plaiting fog'. The school persevered, but Katy left education with a very limited ability in maths. I don't know where she is now, or how she is managing. I hope she is OK.

This experience ignited my interest as I quite simply had never (to my knowledge) met anyone so limited with regard to number. Some individual research and a third-year dissertation on dyscalculia followed and my work is now continuing, a decade later, as doctoral research.

In a recent report by the OECD, England and Northern Ireland have poor numeracy figures compared with the rest of the world. The figures show that England and Northern Ireland currently rank 14th out of 22 countries for numeracy in adults. This has an obvious economic effect on the individual and the country. Currently 17 million adults are working at entry-level mathematics so there is clearly an issue with the lower end of the ability spectrum. This is where my work is focused.

Why does 'the fog' descend?

There are many differing definitions of dyscalculia or mathematics disorder. DSM-5 places it under the umbrella term of Specific Learning Disorder, meaning it is diagnosed when there are persistent difficulties throughout school years and well below average scores that cannot be explained by other factors. Work on the causes of dyscalculia is being done across the world and the work continues.

There are numerous reasons why someone might struggle with mathematics. These include maths anxiety (Ma & Xu, 2004) and issues with short-term memory (Henry & MacLean, 2003), but I am particularly interested in the core cognitive skills involved in mathematics thinking and processing. This is sometimes called 'number sense' and has been researched in various centres around the world.

Although currently still to find its way into a dictionary, number sense (or 'numerosity') is defined as the ability to perceive number. This is believed to be an innate mechanism (Feigenson et al., 2004) that develops throughout childhood and progresses into adulthood. It has even been documented as an ability in ethological terms, with work on apes showing that they have an understanding of number (Gallistel, 2009). The perception of quantity in a visual field is called subitising and it is the ability to perceive the number of dots in an array. Typical development means that an individual can perceive four items in milliseconds without the need to count them: larger arrays take longer but they can still be quick. From this subitising an individual can then develop a sense of number that forms the building blocks of their understanding of mathematics (Lyons & Beilock, 2011). Problems with this ability means individuals can struggle greatly.

This is the case for about 4 per cent of the population: their number sense develops at atypical rates. Figures vary on the prevalence but a large-scale study conducted in Cuba (Reigosa-Crespo et al., 2012) found a prevalence rate of 3.6 per cent. This issue can have profound effects on mathematics learning and, although the problem is by no means rare, many individuals leave the education system unaware of the reason for their poor ability in mathematics.

There are tools available to screen for number sense issues that can be completed online. The online screener I use in my research (tinyurl.com/msg95wa) involves five tasks, all of which are timed. The first is a baseline reaction time test to establish if slow response can be attributed to computation difficulties or slow reaction time. The second task is called the dot enumeration task, requiring the participant to compare an array of dots with a number and stating whether they are of the same magnitude or not. The third is a kind of numerical 'Stroop test', which has two numbers on screen, one in a bigger font size than the other: the screener then asks you to say which is more. The fourth and fifth tasks ask you to say whether addition or multiplication sums presented individually are correct.

The online test takes a maximum of 30 minutes, and a whole class can complete it concurrently. This has advantages in reducing time out of lessons. However, it is not used in many schools and therefore the identification of students with number sense issues does not happen. It should be noted that the screener also only highlights where there might be an issue. No formal diagnosis of dyscalculia can be made. Students like Katy, therefore, can label themselves as just 'bad at maths'. Despite the school's (and often the student's) best efforts, they will leave education with very poor skills. Number sense has been shown to underpin future success in mathematics (Starr et al., 2013) and so by not being identified there will always be
a problem for the individual accessing the curriculum. The fog remains.

Lifting the fog

If there was nothing that could be done for someone with number sense issues then identifying it would be, in some cases, unhelpful. Having a reason behind the problem would help some individuals, but we can go further – intervention can help those with number sense issues improve their mathematical skills (Wilson et al., 2009). The recent work of Panamath (www.panamath.org) on lifespan development of number sense also highlights that there is not a critical period for its development in the early years. It seems, at least in the general sample used of 11,000 participants, to continue to develop into the thirties (Halberda, 2012). This means that there are people who have left school feeling absolutely useless at maths who can still be helped with actually very little intervention time at all.

Most of the work has been conducted on early-years intervention as there is little doubt that is the best time to detect and intervene with the problem. This will avoid the development of maths anxiety through bad experiences, the labelling from teachers and the psychological effects that might occur as a consequence of having poor number sense. Ironically, all these by-products of poor number sense impinge on maths abilities too, so the negative effect is cumulative.

Interventions developed for the early-years/primary age group are available. The Number Sense computer program allows children to develop their number sense link to symbolic mathematics through interaction with a computer (Wilson et al., 2009), allowing a bespoke approach. It also means that it is less labour intensive and therefore cheaper for schools. Despite these interventions, however, there are still children being missed on their transition to secondary school. This is possibly due to lack of awareness in teachers of number sense issues, and sometimes because there is no clear diagnostic test that allows for an educational psychologist to diagnose the condition.

My research

My work is focusing on adolescents, like Katy, who are in the low-ability maths sets and are struggling to grasp a curriculum that assumes number sense has been developed. I started my PhD in January 2013 and am looking at identification of students with number sense issues. Could data already held by schools help to identify those with number sense issues? Early indications suggest not.

I am also examining the psychological effects that having number sense issues, and therefore possibly dyscalculia, can have on an individual. Work on dyslexia highlights the psychological effect that having a learning problem can have (Ingesson, 2007). This might influence policy both within schools and nationally with regard to maths education.

Later in the lifespan intervention strategies become less frequent. Of course, the work done in early years can still be applied, but you run a risk as a practitioner of patronising a teenager or adult with plastic bricks and sticks designed for children under 10. If handled well, these can still help, but it could potentially increase already poor self-esteem in a secondary school student and perpetuate the idea that they are stupid. My work is with secondary-age children because that's where my experience lies and it is an area where much work needs to be done. I still find it shocking that there is so little awareness of this issue at all. In comparison with problems in literacy (which is another kind of 'fog' that individuals have to live with) the research field has much progress to make. That is what this article is about; raising awareness. To many of you the idea of number sense and dyscalculia is not new. To others, through no fault of your own, it is not.

Next time you witness someone with a purse brimming with coins who hands over a note to avoid the embarrassment of counting out the change, or you get frustrated by someone's inability to keep track of time… please remember this article. Yes, they may be lazy or a daydreamer, but they also equally might be finding their way through the 'number fog'.

Eleanor Willard is a part-time lecturer and PhD researcher into difficulties in learning mathematics at Leeds Beckett University
[email protected]

References

Feigenson, L., Dehaene, S. & Spelke, E.S. (2004). Core systems of number. Trends in Cognitive Neuroscience, 8(7), 307–314.
Gallistel, C.R. (2009). Animal cognition: The representation of space, time and number. Annual Review of Psychology, 40, 155–189
Halberda, J., Ly, R., Wilmer, J.B. et al. (2012). Number sense across the lifespan as revealed by a massive internet-based sample. PNAS, 109(28) 11116–11120.
Henry, L. & MacLean, M. (2003). Relationships between working memory, expressive vocabulary and arithmetical reasoning in children with and without intellectual disabilities. Educational and Child Psychology, 20(3), 51–63.
Ingesson, G. (2007). Growing up with dyslexia – Interviews with teenagers and young adults. School Psychology International, 28, 574–591.
Lyons, I.M. & Beilock, S.L. ( 2011). Numerical ordering ability mediates the relation between number-sense and arithmetic competence. Cognition, 121, 256–261.
Ma, X. & Jianming, X. ( 2004). The causal ordering of mathematics anxiety and mathematics achievement: A longitudinal panel analysis. Journal of Adolescence, 27(2), 165–179.
Reigosa-Crespo, V., Butterworth, B.,  Estevez, N. et al. (2012). Basic numerical capacities and prevalence of developmental dyscalculia: The Havana Survey. Developmental Psychology, 48(1), 123–135.
Starr, A., Libertus, M.E. & Brannon, E.M. (2013). Number sense in infancy predicts mathematical abilities in childhood. PNAS, 110(45), 18116–18120.
Wilson, A.J., Dehaene, S., Dubois, O. & Fayol, M. ( 2009). Effects of an adaptive game intervention on accessing number sense in low-socioeconomic-status kindergarten children. Mind Brain and Education, 3(4), 224–234.