# Developmental Origins of Numerical Cognition

Nearly everyone has heard of developmental dyslexia - a learning disorder characterized by poor reading skills despite otherwise sufficient schooling - but have you heard of developmental dyscalculia? Many people have not. Today begins a week-long series on this lesser-known learning disorder. First, we'll consider some potentially innate mechanisms of numerical cognition that give rise to more complex mathematics.

In his 1933 novel Miss Lonelyhearts, Nathanael West writes, “Numbers constitute the only universal language.” Humans have a natural tendency to classify and quantify objects and events around them. Numbers and arithmetic are so basic to the human experience that children develop a basic sense of number and mathematical relations without explicit instruction.

Developmental Building Blocks
In the 1960s, Piaget proposed a three-stage sequence to number acquisition. In stage one, children do not understand one-to-one correspondence of objects – that is, when shown an array of five white jelly beans, they cannot match them to the proper number of black jelly beans. In stage two, an instinctive one-to-one correspondence emerges where children begin to grasp the fundamental idea of equivalence in number, but only if the two sets of objects are equal in all dimensions (number and density, for example). The third stage child understands equivalence more fully, not being fooled by a change in density (i.e. physical proximity of each item in a set) to think that the number of jelly beans has changed.

A stage two child would not consider the two rows of dots as equivalent, since they have different densities.

While Piaget was a great experimentalist, many of his experiments were critically flawed (he experimented on his own children!), and development in general probably doesn't proceed along in a series of stages. More recently, in a more sophisticated series of experiments involving the brief presentation of arrays of dots on a screen, Xu and Spelke demonstrated that six-month-old infants were able to discriminate between eight and sixteen, and between sixteen and thirty two. However, the infants did not discriminate eight dots from twelve or sixteen from twenty four. Starkey and Cooper demonstrated that infants were unable to discriminate four from six dots, in a similar experiment. The findings suggest that infants can discriminate to 2:1 ratios such as 16:8 and 32:16, but not 3:2 ratios such as 12:8 or 6:4. Critically, these dot arrays were presented too quickly for the infants to count them (and even so, infants aren't yet able to explicitly count, since counting requires language); instead, some other mental process was engaged in order to quickly estimate the number of items in the array.

A second set of experiments by Lipton and Spelke sought to determine whether this finding was limited to the visual field, or also applied to auditory input. Infants heard sequences of sounds from a right-side and left-side speaker. The infants were again sensitive to 2:1 ratios (16 and 8 sounds) but not 3:2 ratios (12 and 8 sounds). These findings suggest that representations of approximate numerosities are independent of sensory modality or stimulus format.

In a third set of experiments, Spelke and Xu repeated their dot-array experiments with smaller numbers of dots: arrays of either one versus two dots, or two versus three dots. The findings of these studies indicated that although infants treat large numbers of visible items as a set, they appear to treat small numbers of visible items as individual objects, and not as a set of objects with a cardinal value.

Taken together, these experiments (and many others) suggest that very early in development, infants are able to engage at least two cognitive mechanisms that contribute to numerical cognition: an approximate large number system, and an exact small number system (for up to 3-4 objects).

From Numerosity to Mathematics
Most children eventually acquire four primary mathematical abilities without explicit instruction:
(1) numerosity, which is the ability to determine the quantity of items in a set without counting;
(2) ordinality, which is a basic understanding of "more than" and "less than" relationships between sets of objects;
(3) counting, which is the ability to determine how many items are in a set using a system of symbolic representation – a preverbal counting system has been observed, as well as a language-based system; and
(4) simple arithmetic, which is an understanding of and sensitivity for increases (addition) or decreases (subtraction) from a set. (Geary, 1995)

Unlike basic number abilities, calculation ability represents an extremely complex cognitive process, and requires explicit instruction. The loss of the ability to perform calculation tasks resulting from neuropathology is known as acalculia or acquired dyscalculia, which is an acquired disturbance in computational ability. The developmental defect in the acquisition of numerical abilities, on the other hand, is usually referred to as developmental dyscalculia, or simply dyscalculia.

This was the first in a week-long series on developmental dyscalculia.

Xu, F. (2000). Large number discrimination in 6-month-old infants Cognition, 74 (1) DOI: 10.1016/S0010-0277(99)00066-9

Starkey, P., & Cooper, R. (1980). Perception of numbers by human infants Science, 210 (4473), 1033-1035 DOI: 10.1126/science.7434014

Lipton JS, & Spelke ES (2003). Origins of number sense. Large-number discrimination in human infants. Psychological science : a journal of the American Psychological Society / APS, 14 (5), 396-401 PMID: 12930467

Geary DC (1995). Reflections of evolution and culture in children's cognition. Implications for mathematical development and instruction. The American psychologist, 50 (1), 24-37 PMID: 7872578

• palmd says:

Keep it coming---and keep it archived for your book. You are going to write a book, aren't you?

• Matty Smith says:

As someone with dyscalculia I'm really appreciating this series. Know what bothers me most of all: Analogue clocks.

When you ask someone the time, and they flash their watch at you... ...it's just so embarrassing. I never tell them they haven't helped me a single bit. I just sort of nod and say thanks and go try and find out what the time is some other way. Really, I should just swallow my pride and ask them to read it out for me.

• This is a nice discussion of the analog magnitude system. Are you planning on covering the other half (the development of the absolute magnitude system)? I realize there are those who try to collapse all to analog magnitude, though I personally don't think that's likely to work, and in any case as a theory it at least needs to be discussed.

Here's a three-part series I did on counting: part 1, part 2, part 3.

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• Jason is a graduate student in Developmental Psychology at the University of Southern California. He is interested in the way that the environment interacts with biology in producing innate or skilled behavior, and the evolution of the mind. He is also Psychology and Neuroscience Editor for Research Blogging and Editor of Open Lab 2010.He also blogs about animal behavior and cognition at The Thoughtful Animal.

Melody is a researcher in cognitive science and an occasional contributor to Scientific American's "Mind Matters."