Ordering

Davis and Perusse (1988) argued that making a relative numerosity judgment is simpler than the ability to form an absolute representation of number. The implicit idea here is that it is not necessary to represent cardinal values when making relative numerosity judgments. However, an alternative possibility is that ordering is an operation that is performed on absolute numerical representations. It is likely that there is a continuum of complexity for both cardinal and ordinal judgments.

Honig and colleagues (Honig, 1991; Honig and Matheson, 1995; Honig and Stewart, 1989, 1993) have conducted a series of studies showing that pigeons are sensitive to the relative number of icons in visual matrices. In one study, pigeons were trained to respond to a homogeneous array of X's and to avoid responding to a homogeneous array of O's (Honig and Stewart, 1989). Pigeons were then tested with arrays that had different proportions of X's and O's, and the proportion of responses the pigeons made was a systematic function of the proportion of X's in the test arrays. These data suggest that the pigeons were responding on the basis of the relative numerosity of the X's compared to the O's. Similarly, Meck and Church (1983) trained rats to discriminate two from eight sounds or successively presented visual stimuli and found a generalization gradient for intermediate values (for a similar demonstration in pigeons, see Roberts, 1995). More recently, Emmerton (1998) used a similar paradigm and demonstrated that pigeons trained to discriminate one and two from six and seven simultaneously presented visual stimuli subsequently showed a generalization gradient to the intermediate values.

Many primate species are adept at choosing the larger of two quantities of food (e.g., Anderson et al., 2000; Boysen and Berntson, 1995; Boysen et al., 2001; Call, 2000; Dooley and Gill, 1977; Rumbaugh et al., 1987, 1988). For example, Rumbaugh et al. (1987, 1988) presented chimpanzees with two food wells that contained different numbers of discrete food items and then measured the percentage of trials in which the chimps chose the larger quantity. They found that with values 0 through

4 the chimpanzees chose the larger quantity on approximately 95% of trials. Beran (2001) has also shown that chimpanzees can track sequential accumulations into two separate containers and subsequently compare the two nonvisible quantities to choose the larger quantity. In that study, chimpanzees watched as each of two buckets received up to three smaller sets in alternation (see also Call, 2000). Accuracy in these comparison tasks is controlled by the ratio of the two values being compared (Dooley and Gill, 1977). Interestingly, chimpanzees seem completely unable to learn to choose the smaller of two food quantities in order to obtain the larger quantity, suggesting that chimpanzees may be unable to inhibit their desire to choose the larger quantity (Boysen et al., 1999). Orangutans may differ in this respect (Shu-maker et al., 2001).

Thomas et al. (1980) presented squirrel monkeys with two random dot patterns and reinforced them for choosing the display with the smaller number. The monkeys were trained until they reached a performance criterion on successive numerical pairs beginning with 1 vs. 2. One monkey reached criterion with 7 vs. 8, and a second monkey reached criterion with 8 vs. 9. These data suggest that monkeys may appreciate ordinal relations between numerosities; however, because each numerical pair was trained successively, it remains unclear whether the monkeys learned a series of absolute discriminations or were using a more abstract ordering operation (see also Thomas and Chase, 1980).

Washburn and Rumbaugh (1991) trained rhesus monkeys to use a joystick to select one of two arabic numerals presented on a computer monitor. Whichever numeral the monkey selected resulted in the immediate delivery of the corresponding number of food pellets. The monkeys learned to choose the symbol that produced the larger number of pellets and succeeded on tests of transitive inference. In addition, the monkeys' accuracy was dependent on the numerical distance between the quantities the symbols represented. The monkeys may have represented the discrete number of pellets associated with each numeral; alternatively, they may have represented the total amount of food each numeral yielded. Regardless, these results show an impressive capacity to order symbols based on underlying magnitudes (for a similar demonstration with squirrel monkeys, see Olthof et al., 1997; with dolphins, Mitchell et al., 1985).

In a series of studies, Brannon and Terrace (1998, 1999, 2000, 2002; Brannon et al., unpublished data) have examined the ability of rhesus macaques to discriminate the numerosities 1 to 9 and represent their ordinal relations. In these studies rhesus monkeys were first trained to respond in ascending or descending order to a small set of numerical values (1-2-3-4; 4-3-2-1). Nonnumerical confounds were controlled by varying relative element size over a large number of stimulus sets (see Figure 6.4). After extensive training, the monkeys were tested on their ability to order novel exemplars of these same numerosities where nonnumerical dimensions were varied randomly across sets of exemplars. The monkeys learned the response rule and performed well above chance when tested with novel exemplars of the numerosities, regardless of the ordinal direction used in training.

However, were the monkeys using an ordinal numerical rule? Did their behavior reveal any appreciation of the fact that 2 is more than 1, or that 3 is less than 4? To address this question, the monkeys were then tested on their ability to order novel a)

Equal Size

Equal Surface Area Random Size

Clip Art b)

Clip Art Mixed #

Random Size and Shape

Random Size, Shape and Color

Smaller Numerosity Has:

Larger Area Smaller Area

FIGURE 6.4 (a) Exemplars of the seven different types of stimulus sets used by Brannon and Terrace (1998). Equal size: Elements were of same size and shape. Equal area: Cumulative area of elements was equal. Random size: Element size varied randomly across stimuli. Clip art: Identical nongeometric elements selected from clip art software. Clip art mixed: Clip art elements of variable shape. Random size and shape: Elements within a stimulus were varied randomly in size and shape. Random size, shape, and color: Same as previous, with background and foreground colors varied between stimuli. (b) Examples of stimulus sets used in the pair-wise numerosity test. (c) Examples of pair-wise tests where the smaller numerosity has the larger and smaller surface areas.

numerical values that were outside the range of the training values (e.g., 5 vs. 8). The size of the elements was varied such that the smaller numerosity had the larger surface area on 50% of trials. If the monkeys had learned a specific arbitrary ordering of a set of numerical values, then they should have had no basis for ordering novel values that exceed the training range. To illustrate that point, imagine having learned the beginning of an alphabet and then being asked to order the letters at the end of the alphabet. This would be an impossible task given that the ordering of an alphabet is arbitrary. If, on the other hand, the monkeys learned an ordinal rule such as "respond to the stimulus with the smallest number of elements and continue to do so without replacement," then they should have had no trouble ordering pairs of novel values.

Rosencrantz and Macduff, trained to respond in ascending order to the numer-osities 1 to 4, performed at above-chance levels when tested with pairs of the novel values 5 to 9. These results show that monkeys do not require training on each specific numerical pair to appreciate their ordinal relations. In fact, the monkeys were equally good at this task when the smaller number had a larger surface area compared to when it had a smaller surface area. These data have recently been replicated in a baboon, a squirrel monkey, and two cebus monkeys (Evans et al., 2002; Smith et al., 2002). In contrast to Rosencrantz and Macduff's success in extrapolating the ascending numerical rule to novel values, Benedict, the monkey trained to respond in descending order to the values 1 to 4 (Brannon and Terrace, 2000), was at chance when tested with novel numerosities.

We have recently conducted a new experiment that sheds light on Benedict's failure to generalize to novel values after descending training on the values 1 to 4 (Brannon et al., unpublished data). In this experiment we trained monkeys to respond 4-5-6 or 6-5-4 and then tested their ability to order pairs of all the values 1 to 9. The monkeys learned the 4-5-6 and 6-5-4 rule and were able to appropriately order novel exemplars of 4 to 6 with above-chance performance. However, when tested with pairs of the values 1 to 9, the monkeys showed an interaction whereby they were above chance when tested with values that continued the training sequence (e.g., 4-5-6 training yielded good 7-8-9 pairwise performance, and 6-5-4 training yielded good 3-2-1 pairwise performance), but below chance when tested with values that preceded the training range (Brannon et al., unpublished data). These results suggest two important conclusions. First, familiarity and novelty are not the most important factors in monkeys' numerical discriminations. In fact, the monkeys were in some cases able to order novel pairs with higher accuracy than familiar pairs. Second, in addition to learning about the ordinal direction in which they are expected to respond, monkeys seem to learn something more specific about the anchor point in the training series. The first value learned in the training sequence seems to act as an anchor and figure heavily into the monkeys subsequent comparison calculations. To illustrate, consider the 4-5-6 training series and the test pairs 1 vs. 3 and 7 vs. 9. The value 3 is closer than 1 to the anchor value 4, whereas the value 7 is closer than 9 to the anchor value 4. Thus, the monkey will successfully order 7 and then 9, but unsuccessfully order 3 and then 1.

Moyer and Landauer (1967) first showed that when adults are presented with two arabic numerals and required to indicate the symbol that represents the larger magnitude, their accuracy and latency are systematically influenced by the numerical disparity between the magnitudes. Specifically, reaction time decreases and accuracy increases with increasing numerical disparity. Furthermore, when distance is held constant, performance decreases with increasing numerical magnitude. Since this pioneering work, the distance and magnitude effects have been replicated in many languages, and representational formats with adult humans and have also been found in children as young as 5 years old (e.g., Buckley and Gilman, 1974; Temple and Posner, 1998).

Moyer and Landauer (1967) interpreted their results as evidence that numbers are represented as analog magnitudes. This easily replicated finding is also a signature property of animal number discrimination. For example, Brannon and Terrace (1998, 2000) showed that when rhesus monkeys choose the larger or smaller of two visual quantities, their reaction time and accuracy are systematically affected by the numerical disparity between the response alternatives and their magnitudes. Thus, number discrimination follows Weber's law such that as absolute magnitude increases, a larger disparity is needed to obtain the same level of discrimination. Brannon and Terrace then tested college students with the same task and stimuli and instructed the participants to respond as quickly as they could while still getting the majority of problems correct. Figure 6.5 shows the results for college students and monkeys. The striking similarity in the latency and accuracy functions obtained with

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