The last installment of “Alex’s Number Series”—and maybe one of the most interesting, because here I discuss some numerical abilities that have not yet been demonstrated in any nonhuman other than Alex! Such is not to say that it won’t or can’t be demonstrated in another nonhuman, only that Alex was the first and so far the only nonhuman to show this behavior. The technical description of this behavior is that he inferred the cardinal values of new numbers from their ordinal values.
In layman’s terms, what he did was to figure out that the value of every number on his number line (a list of numbers in order: “one, two, three…etc.”) was one more than the number before it, and one less than the number after it (formally known as the “successor function”) and, after learning two new labels for his number line, inferred the numerical value of those labels based on their position on that number line.
The study began when I started a position at the Radcliffe Institute at Harvard in 2004 and initiated a collaboration with two colleagues, Susan Carey and Elizabeth Spelke, who were studying number concepts in children. They both thought that nonhumans really didn’t have the same understanding of numbers as did children, and were eager to test out that hypothesis.
Toddler Talk: Stringing Numbers Together
The issues were as follows. Young children learn their small numbers very slowly (reviewed in Carey, 2009): Not until they are about 2½ years old do they comprehend the concept of “one versus many.” It takes about another nine months for them to understand “one versus two versus many,” and about another five months to succeed on “one versus two versus three versus many.”
Around that time they also learn to recite a number line. Initially, they just string together labels they know are associated with numbers in random patterns (“one, three, six, two”; Fuson, 1988; Le Corre, Van de Walle, Brannon, & Carey, 2006). As they learn more about the meanings of these labels, they begin to understand that the numbers in the line have a particular order (“one, two, three…”; see review in Thompson & Siegler, 2010), and about the time they understand the meaning of “four,” they make the connection between the labels in the number line and the labels that represent quantity, and infer the successor function described above…they immediately comprehend the meaning of the labels “five”, “six”, etc. without any formal instruction (Carey, 2009; Hurtford, 1987).
Alex Gets Vocal About Numbers
Nonhumans, however, take just as long to learn their larger numbers as their smaller ones (e.g., Biron & Matsuzawa, 2001). One problem is that nonhumans are never taught to produce a number line but, as we saw in previous entries here, they can learn the order of their numerals. But what about Alex? Unlike the apes, he hadn’t needed training to understand the order of his numerals; might Alex’s difference from children in terms of acquiring the larger number labels simply involve difficulty in acquiring the English sounds needed to produce the number labels? For any English label, Alex had to learn to coordinate his syrinx, tracheal muscles, glottis, larynx, tongue height and protrusion, beak opening, and even esophagus (Patterson & Pepperberg, 1998). Separating the issues of vocal versus conceptual learning might test this possibility.
In order to determine if production of English speech was the limiting step in Alex’s ability to learn about new numerical labels, we designed the following experiment (Pepperberg & Carey, 2012): We taught Alex to vocally label the Arabic numerals 7 and 8 in the absence of their respective quantities (i.e., any sets of objects). We then trained him that 6<7<8, using our modeling system and plastic numerals, then tested how 7 and 8 related to his other Arabic labels, the same way we had tested his original understanding of the order of his numbers (Pepperberg & Gordon, 2005). If he inferred the new complete number line, he could then be tested on whether he, like children (≥4 years old), spontaneously understood that “seven” represented exactly one more than “sih”, that “eight” represented two more than “sih” and one more than “seven”, by labeling appropriate physical sets on first trials.
Remember, all he knew at this point was that these numerals represented something greater than 6…they could have represented ten or twenty! Data already showed he knew “sih” represented exactly, not approximately, six items. If he succeeded on the new labels, we could claim that he had figured out, without any training whatsoever, the actual (i.e., “cardinal”) meanings of “seven” and “eight” from their positions on his number line (i.e., from their “ordinality”), something no ape (although evolutionarily closer to humans) had yet achieved.
Alex learned to vocally produce the novel Arabic numeral labels; he then figured out exactly where they belonged appropriately in his number line. But, again, he had learned only that they were greater than 6… yet he appropriately labeled, on first trials, novel sets of seven and eight physical items—he did not have to be taught the relationship between the labels and the novel sets (Pepperberg & Carey, 2012). Thus, he responded exactly as would children who knew their counting principles!
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