A Model of Exact Small-Number Representation

Article excerpt

To account for the size effect in numerical comparison, three assumptions about the internal structure of the mental number line (e.g., Dehaene, 1992) have been proposed. These are magnitude coding (e.g., Zorzi & Butterworth, 1999), compressed scaling (e.g., Dehaene, 1992), and increasing variability (e.g., Gallistel & Gelman, 1992). However, there are other tasks besides numerical comparison for which there is clear evidence that the mental number line is accessed, and no size effect has been observed in these tasks. This is contrary to the predictions of these three assumptions. Moreover, all three assumptions have difficulties explaining certain symmetries in priming studies of number naming and parity judgment. We propose a neural network model that avoids these three assumptions but, instead, uses place coding, linear scaling, and constant variability on the mental number line. We train the model on naming, parity judgment, and comparison and show that the size effect appears in comparison, but not in naming or parity judgment. Moreover, no asymmetries appear in primed naming or primed parity judgment with this model, in line with empirical data. Implications of our findings are discussed.

How are numerical values mentally represented? The obvious way in which to investigate this issue behaviorally is to use tasks that rely critically on number magnitude, the most straightforward task being numerical comparison. This approach dates back at least to Moyer and Landauer's (1967) seminal article that described two critical factors determining number comparison performance: the distance effect and the size effect. Most current theories are rooted in this work.

The distance effect reflects the fact that comparison times are shorter for a larger numerical distance between two numbers that have to be compared. For example, comparison is faster for 2 and 6 than for 2 and 3. The second factor is the size of the numbers: For a given distance, comparison is faster for smaller numbers (e.g., 2 and 4) than for larger numbers (e.g., 7 and 9). The distance and size effects affect not only processing speed but also accuracy. The size and distance effects are robust phenomena and occur with numbers presented in various formats-for example, Arabic notation (Banks, Fujii, & Kayra-Stuart, 1976; Dehaene, Dupoux, & Mehler, 1990; Sekuler, Rubin, & Armstrong, 1971), verbal notation (Koechlin, Naccache, Block, & Dehaene, 1999), and nonsymbolic notation, such as collections of dots (Buckley & Gillman, 1974).

Together, the two effects put crucial constraints on how number magnitude is represented and processed in various tasks. A generally accepted idea is that mental number representations can be seen as organized along a mental number line-that is, a set of units in which close numbers are represented with overlapping distributions of activation. This number line assumption can explain the distance effect, because numbers close to each other (e.g., 1 and 2) have more distributional overlap than do numbers that are more distant (e.g., 1 and 4), and it will be more difficult to discriminate the closer numbers.

To account for the size effect, additional assumptions are needed. One assumption that makes it possible to explain the size effect is magnitude coding. This means that the mental code for a number is analogous to the magnitude it represents. For instance, if a given number activates a set of units on the mental number line, this set of activated units is a subset of the units activated for a larger number (e.g., Zorzi & Butterworth, 1999; see Figure 1A for a graphical representation). With magnitude coding, number comparison is similar to physical magnitude discrimination, which is robustly characterized by distance and size effects, as has long been known from psychophysics (e.g., Weber's law; Festinger, 1943). Another possible assumption that makes it possible to account for the size effect is compressed scaling (e. …


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