## Brief Description Of Five Models Of Nonverbal Number Representation

The sections above describe what nonhuman animals and human infants are capable of in the numerical domain. But how are they accomplishing these mathematical feats? What is the process by which they form numerical representations, and how can we best describe the format of their numerical representations? In this section we will describe five proposals for how number is represented and then review the evidence that supports each model.

### 6.6.1 Arbitrary Numeron Hypothesis

The first proposal is that animals and preverbal children possess an ordered set of arbitrary nonverbal neural representations of numerosities (Gelman and Gallistel, 1978). In the original formulation of this model, numerons were described as indirect symbols with an arbitrary relationship to the numerosities they represent (see Figure 6.10a). Like words or arabic numerals, the proposal was that numerons did not in any way resemble the numerosities they served to represent. Numerons were applied to the set of objects or events through a process conforming to Gelman and Gallistel's five counting principles (Gelman and Gallistel, 1978). First, the one-to-one principle states that each element in the to-be-counted set must be mapped to one and only one numeron (e.g., the verbal label three can be applied to only one of the items in a set). Second, the stable-order principle requires that numerons be applied in a consistent order (e.g., one cannot count a five-element sequence as 1-

FIGURE 6.10 Format of number representations for the values 1, 3, and 5. (a) Arbitrary numeron model: Each number is represented by an abstract, arbitrary symbol. (b) Object-file model: Each object is represented by an object-file. There is no symbol that represents the set of objects. System is limited by the small number of available object-files. (c) Mode-control model: Number is represented as the accrual of pulses from a pacemaker into an accumulator. (d) Connectionist timing model: Memory for intervals is represented as a storage vector that contains information about the half phases of multiple oscillators. Number is derived by dividing the total time that has elapsed by the duration of the intervals between events. (e) Dehaene and Changeux neural network model: Number is represented by a map of numerosity detectors.

FIGURE 6.10 Format of number representations for the values 1, 3, and 5. (a) Arbitrary numeron model: Each number is represented by an abstract, arbitrary symbol. (b) Object-file model: Each object is represented by an object-file. There is no symbol that represents the set of objects. System is limited by the small number of available object-files. (c) Mode-control model: Number is represented as the accrual of pulses from a pacemaker into an accumulator. (d) Connectionist timing model: Memory for intervals is represented as a storage vector that contains information about the half phases of multiple oscillators. Number is derived by dividing the total time that has elapsed by the duration of the intervals between events. (e) Dehaene and Changeux neural network model: Number is represented by a map of numerosity detectors.

2-3-4-5 on one occasion and 1-3-4-5-2 on the next). The third principle is cardinality; it states that the last numeron assigned to an element of a set must also serve to represent the whole set. The final two principles are not essential to the counting process, but describe counting in its most abstract sense. The fourth principle is the abstraction principle; it states that anything can be counted (apples, people, buildings, sounds, etc.). The final principle is the order irrelevance principle; it states that objects can be counted in any order (e.g., left to right, circular pattern, etc.).

### 6.6.2 Subitizing and the Object-File Mode

A popular distinction in the literature of human and animal numerical abilities is subitizing vs. counting. Subitizing is defined as a fast, effortless, parallel perceptual process that is limited to the apprehension of the small values 1 to 4 (Kaufman et al., 1949; Mandler and Shebo, 1982). In contrast, counting is a slower, more effortful sequential process that is used for sets of five or more items.

More recently, the subitizing hypothesis has been further specified in the object-file model, which posits that an object-file is opened for each element in a visual array (Carey, 1998; Hauser and Carey, 1998; Leslie et al., 1998; Simon, 1997; Uller et al., 1999). In this model there is no symbol that represents the numerosity of a set; instead, as shown in Figure 6.10b, each element is represented by a file stripped of object features such as color, shape, and size (see also Trick and Pylyshyn's (1993, 1994) application of the FINST model). The model posits that the visual system contains a limited number of object-files that can be assigned to an object. A given set can only be represented if there are a sufficient number of object-files available (Pylyshyn and Storm, 1988). Thus, if an organism's only means of representing number was the object-file model, it could only represent small sets of objects.

### 6.6.3 Mode-Control Model

The third proposal is the mode-control model, or accumulator model; it shares with the arbitrary numeron hypothesis the idea that animals use a serial process that conforms to the counting definition proposed by Gelman and Gallistel (1978). However, the mode-control model (Meck and Church, 1983; Meck et al., 1985) posits that number is represented as continuous magnitudes that directly reflect the magnitude of the discrete quantities they serve to represent (see Figure 6.10c). Thus the mode-control model contends that the nervous system inverts the representational convention whereby numbers are used to represent linear magnitudes. It is proposed that instead of using number to represent magnitude, the nervous system uses magnitude to represent number (see Gallistel and Gelman, 2000; Meck, 1997). Furthermore, the mode-control model posits that a single mechanism serves to represent time and number.

The mode-control model was originally developed as an adaptation of the information-processing model of animal timing behavior by Gibbon and Church (1984). Like the pure timing model, the mode-control model is composed of a pacemaker, accumulator, working memory buffer, reference memory, and comparator (see Figure 6.11a). At the onset of a relevant stimulus, pulses are gated into an accumulator, which then integrates the number of pulses over time. The critical innovation for the mode-control model is the addition of a mode switch, which allows the system to work like a timer or a counter. Pulses are gated into the accumulator by one of three different modes, depending on the nature of the stimulus. These three modes provide the mechanism with the ability to act as both a counter and a timer. In the run mode, the initial stimulus starts an accumulation process that continues until the end of the signal or trial; in the stop mode, the process occurs whenever the stimulus is physically present; in the event mode, each onset of the stimulus produces a relatively fixed duration of the process regardless of stimulus duration. This mechanism thus provides a way to estimate duration (the run and stop modes) or number (the event mode). The accumulator value is transferred to working memory and sometimes to reference memory. The animal then compares the current value in the

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