“Symbolic numbers are a uniquely human, cultural invention”
Developmental cognitive neuroscientist Daniel Ansari explains how children develop number abilities, and why some people have better numerical and mathematical abilities than others.
Meeri Kim: The ability to count precedes the ability to enumerate, or to determine the total number of items in a set, which is an interesting distinction. By what process do children learn these concepts?
Daniel Ansari: At the heart of basic number development lies the ability to be able to enumerate sets. Enumerating sets typically occurs through counting, and as you already alluded to, children initially learn to count sequences. In other words, early on counting is almost like a meaningless sing-song to young children. They learn the sequence of count words and often eagerly recite it, but they haven’t yet connected them to quantity.
Between three-and-a-half and five years old, children develop an understanding that when you count, you enumerate sets. So when you climb up the stairs and count “1-2-3-4-5,” you’ve just counted a total of five steps. At this point, children have acquired the cardinality principle — cardinality being the total number of items in a set. This is a major milestone in the development of early number abilities because now they have a symbolic representation of number. They’re able to appreciate that the number five stands for any sets of five.
Of course, that’s not where it ends. Children also go beyond just having a verbal symbolic representation of numbers to having other symbolic representations of numbers, such as Arabic numerals. And children learn to compare and order sets using symbolic representations. Arabic numerals are very powerful representations of numbers because they allow people across different countries to communicate about numerical magnitude and numerical sets without speaking the same language, and they also allow us to represent large, multidigit numbers using the place value system.
MK: Often you hear people say, “I’m just not a math person” or “I’ve always been terrible at math.” What makes numerical and mathematical skills harder to grasp for some individuals as opposed to others? Does it have to do with failure to learn at an early age?
DA: We know that mathematics achievement is heritable. That does not mean that our mathematical abilities are genetically determined, but it does mean that a significant proportion of the variability (differences between individuals) in children’s and adults’ math ability can be explained by their genes. When we look at the cognitive level, which is what my lab studies, we see that children who struggle with basic calculation often have difficulties with more fundamental low-level skills.
“Early on counting is almost like a meaningless sing-song to young children.”
For instance, they have trouble being able to relate symbols to quantity as well as understanding that five is part of an ordered sequence (e.g. that five comes after four but before six and eight). Some of these building blocks do not seem to be in place for children who go on to experience mathematical learning difficulties.
One of the things that my lab – the Numerical Cognition Laboratory at Western University – is very interested in is identifying and better characterizing what those early building blocks are, how we can screen for children who might have weaknesses in some of these foundational skills, and how we can use that knowledge to inform early intervention strategies. Compared with screeners for early literacy abilities, there are currently very few evidence-based screeners for numeracy in the early years.
Like reading, math is very cumulative. Low-level skills provide the foundation for high-level skills, and those high-level skills then become the foundation for yet higher-level skills. So you can see that if you lack a level of foundational skills, that’s going to impact your learning ability and trajectory going forward.
MK: Describe your work in developmental dyscalculia, a specific learning disability affecting the normal acquisition of arithmetic skills.
DA: Children with developmental dyscalculia are those who have a specific and persistent weakness in the domain of mathematics. In other words, their performance on standardized tests of math is lower than, say, their performance on standardized tests of IQ.
“We currently have no evidence that the brains of children with developmental dyscalculia are qualitatively different from children in a normal range of math achievement.”
These children experience particular difficulties with symbolic representations. They may not struggle so much with being able to discriminate which of two dot arrays is larger, but when it comes to being able to discriminate which of two Arabic numerals is larger, they’re significantly slower and more error-prone than children with relatively normal mathematics achievement.
We also see that they show differences in brain function and structure in regions that are typically associated with number processing and mathematics. But we currently have no evidence that the brains of children with developmental dyscalculia are qualitatively different from children in a normal range of math achievement. That is, the brain regions that show reduced activation in dyscalculia are also those that are involved in number processing among children and adults without mathematical difficulties.
MK: Another facet of your research deals with the structure and circuitry of the brain regions involved in mathematical processing. What are some of your findings in this area?
DA: We do a lot of work in the lab trying to better understand the brain circuits that are engaged when you perform basic numerical tasks. One pattern of data that is very consistent across studies is that symbolic number processing — so even passively viewing number symbols (such as Arabic numerals) when you’re lying in an MRI scanner — is associated with activation of the left parietal cortex. We just published two meta-analyses that demonstrate this very nicely.
“The beauty of studying symbolic numbers is that we’re not born with the ability to represent them – they are a uniquely human, cultural invention.”
The beauty of studying symbolic numbers is that we’re not born with the ability to represent them – they are a uniquely human, cultural invention. So we can study how such symbols become represented in the brain over the course of learning and development. Numerical symbols are something that children need to learn, and therefore they need to construct brain circuits that are eventually correlated with symbolic number processing.
We’re very interested in that learning process, and going forward, we want to do more developmental studies – for instance, longitudinal studies in which we are able to track the neural changes associated with the development of symbolic number processing in children with varying levels of mathematics ability.
Mathematics is fascinating because there are so many different facets to mathematics learning. Learning whole numbers, then going on from learning whole numbers to fractions, and so forth. The upper end of complexity is very high in mathematics, and most of us don’t get to that level where mathematicians are. But at the very heart of it lies this initial step of developing a symbolic representation of numbers and being able to flexibly use them in multiple numerical contexts.
Daniel Ansari is a Professor and Canada Research Chair in Developmental Cognitive Neuroscience in the Department of Psychology and the Brain & Mind Institute at Western University in London, Ontario, where he heads the Numerical Cognition Laboratory. Ansari and his team explore the developmental trajectory underlying both the typical and atypical development of numerical and mathematical skills, using both behavioral and neuroimaging methods. Daniel Ansari is a Jacobs Research Fellow.