There is more to numerical knowledge than the ability to distinguish different numbers. The ability to distinguish numbers does not entail an ability to reason about those numbers, to determine, for example, that five is larger than three or that two is composed of one and one. To determine such relationships, the animal or infant must not only be able to construct mental representations of the relevant numbers but be able to manipulate these representations in numerically meaningful ways. There is empirical evidence that infants and animals can determine the results of certain numerical operations on small numbers of physical objects.
Studies in the author's lab have investigated human infants' numerical reasoning capacities. In one experiment, 5-month-old infants were divided into two groups. Those in the "1 + 1" group were shown a single item being placed into an empty display area. Then a small screen rotated up, hiding the item from view, and the experimenter brought a second identical item into the display area in clear view of the infant. The experimenter then placed the second item out of the infant's sight behind the screen (this sequence of events is shown in the top portion of Fig. 4). Thus, infants could see the nature of the arithmetical operation being performed but could not see the result of the operation. The screen was then dropped to reveal an outcome of either one (the impossible outcome) or two (the possible outcome) objects. Infants in the "2—1" group were similarly presented with a sequence of events depicting a subtraction of one item from two items (shown in the bottom portion of Fig. 4). Again, after this sequence of events was concluded, the screen rotated downward to reveal either one (now the possible outcome) or two (impossible outcome) items in the display case.
Infants' looking time at the display was recorded when the screen dropped. The prediction was that infants would be surprised by an apparently impossible result. Thus, the two groups should show significantly different looking patterns; infants in the 1+1 group should look longer when the result is one than when it is two, in comparison to the 2—1 group, which should show the reverse pattern. (A pair of pretest trials, in which infants were simply shown displays of one and two items, revealed that infants in the two groups did not differ in their baseline looking patterns at one and two items.) This was, in fact, the pattern of results obtained; infants in the two groups differed significantly in their patterns of looking in the test trials. Infants in the 1+1 group looked longer when the addition appeared to result in a single item than when it resulted in two items, whereas infants in the 2—1 group looked longer when the subtraction appeared to result in two items than when it resulted in a single item. To ensure that infants were determining the exact result of the operation (that is, that they were expecting two objects in the 1+1 situation and one object in the 2—1 situation) rather than simply
expecting the number to have been changed in some way as a result of the operation without having expectations as to precisely what the result should be, another experiment was conducted. Here, infants were shown an addition of 1 + 1, with outcomes of two and three objects. In this case, both outcomes are different from the initial number of objects (one) placed in the display. Thus, if infants only expect some change to obtain as a result of the addition, they will not be surprised by either outcome. However, if they are computing the precise nature of the numerical change, they will be expecting two objects behind the screen and will look longer at the incorrect result of three objects. (A pretest condition showed that infants looked equally long at two and three objects.) This was the pattern of results obtained: infants looked longer when the addition appeared to result in three items than in two items.
The studies reviewed previously and others like them show that infants as young as 5 months of age are sensitive to the numerical relationships between small numbers and are able to determine the results of simple numerical operations. Similar abilities have also been shown in somewhat older infants by a task that required them to perform motor actions based on their numerical expectations. In this study 18- to 35-month-olds saw from one to five identically colored ping-pong balls placed into an opaque box and then saw an experimenter either add or remove a small number of balls. They then were allowed to reach into the box to retrieve the objects. The box was constructed so that infants could not see into the box as they were reaching into it, so that each reach into it allowed contact with, and removal of, only one object at a time. Thus, the number of reaches into the box that the infants made indicated how many items they believed the box to contain. Even the 18-month-olds showed a knowledge of how many objects the box contained when small numbers were involved.
Some nonhuman species are also able to compute results of certain numerical operations. The most conclusive evidence comes from studies conducted by Sarah Boysen and Gary Berntson, who taught a female chimpanzee to associate the Arabic numerals 0-4 with their respective numerosities. Without further training, she was able to choose the numeral representing the sum of oranges hidden in any two of the three possible hiding places in her pen. Most impressive of all, when the sets of oranges in the hiding places were replaced with Arabic numerals (one card with an Arabic numeral printed on it hidden in each of any two hiding places), she was immediately able to choose the Arabic numeral representing the sum of the two found numerals. That is, without training, she was able to operate over two symbols representing numerosities in such a way as to arrive at the symbol representing their sum. In another study, free-range wild rhesus monkeys were given a version of the 1+1 addition situation described earlier for human infants. When shown one eggplant and then another eggplant placed in a box out of sight, the moneys looked significantly longer when only one eggplant was revealed than when two were, despite showing no preference in a control situation for looking at one eggplant over two.
Suggestive findings have been obtained with other species as well. Rats appear to anticipate when they are approaching the required number of presses when they must press a required minimum number of times on a lever to obtain a reward. Rats will frequently check for the reward before they have given the required number of presses in situations when there is no penalty for checking for the reward too early and, upon finding no reward, will return to the lever to increase their number of presses. Interestingly, the greater their number of presses before pausing to check for the reward, the smaller their number of additional presses upon returning to the lever. That is, they appear to know how close they are to the needed number, not only whether they have or have not reached that number yet. Finally, in one study, rats were trained to press a lever on the left when presented with either two sounds or two light flashes and a lever on the right when presented with four sounds or four light flashes. Following training, rats were presented with two sound-light flash pairings (a sound accompanied by a simultaneous light flash, followed by another sound accompanied by a light flash). In this situation, rats pressed the right-hand lever, showing that they had computed that there were four stimuli altogether. They were either adding the number of sounds to the number of light flashes or enumerating the stimuli independently of their kind.
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