Symbolic Fractions Elicit an Analog Magnitude Representation in School-age Children

A fundamental question about fractions is whether they are grounded in an abstract non-symbolic magnitude code, similar to those postulated for whole numbers. Mounting evidence suggests that symbolic fractions could be grounded in mechanisms for perceiving non-symbolic ratio magnitudes. However, systematic examination of such mechanisms in children has been lacking. We asked second and fifth grade children (prior to and after formal instructions with fractions, respectively) to compare pairs of symbolic fractions, non-symbolic ratios and mixed symbolic/non-symbolic pairs. This paradigm allowed us to test three key questions: 1) whether children show an analog magnitude code for rational numbers, 2) whether that code is compatible with mental representations of symbolic fractions, and 3) how formal education with fractions affects the symbolic-non-symbolic relation. We examined distance effects as a marker of analog ratio magnitude processing and notation effects as a marker of converting across numerical codes. Second and fifth grade children’s response times and error rates showed classic distance and notation effects. Non-symbolic ratios were processed most efficiently, with mixed and symbolic notations being relatively slower. Children with more formal instruction in symbolic fractions had a significant advantage in comparing symbolic fractions, but a smaller advantage for non-symbolic ratio stimuli. Supplemental analyses showed that second graders relied on numerator distance more than holistic distance, and fifth graders relied on holistic fraction magnitude distance more than numerator distance. These results suggest that children have a non-symbolic ratio magnitude code, and that symbolic fractions can be translated into that magnitude code.

Encoding of numerical information in memory: Magnitude or nominal?

In studies of long-term memory of multi-digit numbers the leading digit tends to be recalled correctly more often than less significant digits, which has been interpreted as evidence for an analog magnitude encoding of the numbers. However, upon closer examination of data from one of these studies we found that the distribution of recall errors does not fit a model based on analog encoding. Rather, the data suggested an alternative hypothesis that each digit of a number is encoded separately in long-term memory, and that encoding of one or more digits sometimes fails due to insufficient attention in which case they are simply guessed when recall is requested, with no regard for the presented value. To test this hypothesis of nominal encoding with value-independent mistakes, we conducted two studies with a total of 1,080 adults who were asked to recall a single piece of numerical information that had been presented in a story they had read earlier. The information was a three-digit number, manipulated between subjects with respect to its value (between 193 and 975), format (Arabic digits or words), and what it counted (baseball caps or grains of sand). Results were consistent with our hypothesis. Further, the leading digit was recalled correctly more often than less significant digits when the number was presented in Arabic digits but not when the number was presented in words; our interpretation of this finding is that the latter format does not focus readers’ attention on the leading digit.

Numerical Values Leave a Semantic Imprint on Associated Signs in Monkeys

Animals and humans share an evolutionary ancient quantity representation which is characterized by analog magnitude features: Discriminating magnitudes becomes more difficult with increasing set sizes (size effect) and with decreasing distance between two numerosities (distance effect). Humans show these effects even with number symbols. We wondered whether monkeys would show the same psychophysical effects with numerical signs and addressed this issue by training three monkeys to associate visual shapes with numerosities. We then confronted the monkeys with trials in which they had to match these visual signs with each other. The monkeys' performance in this shape versus shape protocol was positively correlated with the numerical distance and the magnitudes associated with the signs. Additionally, the monkeys responded significantly slower for signs with higher assigned numerical values. These findings suggest that the numerical values imprint their analog magnitudes characteristics onto the associated visual sign in monkeys, an effect that we also found reflected in the discharges of prefrontal neurons. This provides evidence for a precursor of the human number symbol knowledge.

Symbolic, numeric, and magnitude representations in the parietal cortex

We concur with Cohen Kadosh & Walsh (CK&W) that representation of numbers in the parietal cortex is format dependent. In addition, we suggest that all formats do not automatically, and equally, access analog magnitude representation in the intraparietal sulcus (IPS). Understanding how development, learning, and context lead to differential access of analog magnitude representation is a key question for future research.

On the Perceptual Generality of the Unit-Decade Compatibility Effect

Number magnitude is assumed to be holistically represented along a single mental number line. Recently, we have observed a unit-decade-compatibility effect which is inconsistent with that assumption (Nuerk, Weger, & Willmes, 2001) . In two-digit Arabic number comparison, we have demonstrated that compatible comparisons in which separate decade and unit comparisons lead to the same decision (32_47, 3 < 4 and 2 < 7) were faster than incompatible trials (37_52, 3 < 5, but 7 > 2). Because overall distance was matched, a holistic model could not account for the compatibility effect. However, one could argue that the compatibility effect was due to the specific vertical perceptual arrangement of the two-digit numbers in Nuerk et al.’s (2001) experiment where the decade digits and unit digits were presented column-wise above each other. To examine this objection, we studied the perceptual generality of the compatibility effect with diagonal presentation. We replicated the compatibility effect with diagonal presentation. It is concluded that the compatibility effect is not due to encoding characteristics imposed by the perceptual setting of the original experiment. In particular, the assumption of an overall analog magnitude representation for two-digit numbers is not consistent with these data.