This RAPID project focuses on "fundamental" research on three distinct aspects of number sense: (1) a small exact number system; (2) a large approximate number system; and (3) a system for set-based quantification. Recent exploratory research suggests a strong linkage between number sense (particularly large numerical approximation systems) and student abilities in other domains of mathematics. The investigators will research the links among these three aspects, and the study is designed to further the preliminary findings (cited above).
The investigators propose a correlational study in which they test a population of the indigenous Piraha people of Brazil (a small, isolated, monolingual hunter-gatherer group from the Amazonas) and a sample of Americans (60 in each group) in a battery of cognitively-oriented tasks which measure different core numerical systems as well as other basic cognitive abilities like short-term memory and face perception (as control tasks). The Piraha are an ideal test case for understanding the relationship among core numerical systems because their language is has no words for numbers. In addition, the Piraha do not use exact number in their society and they do not adopt cultural or linguistic conventions from other cultures. A RAPID is justified because their population is threatened by imminent development.
This research is important because a deeper understanding of the conceptual/cognitive components of number sense and how they interrelate can lead to perhaps changed understandings of how students learn and teachers teach this area of mathematics. And, because number sense is so foundational to mathematics and because preliminary research results show its potential importance to future mathematics learning, the project may have a transformative impact on the field.
A fundamental general question regarding numeracy is how it is that children learn to count. In industrialized countries, it is known that children go through a reliable learning progression: first learning the meaning of "1", then the meaning of "2", then the meaning of "3", and sometimes followed by the meaning of "4", but then proceeding to understanding the full recursive count list immediately after this (Wynn, 1992). Following the terminology in the literature, we refer to the knowledge of the recursive count list as knowledge of the cardinal principle (CP). Critically, there are stages of learning where children are "1-knowers", "2-knowers", "3-knowers", sometimes "4-knowers" and then "CP-knowers". But children never seem to be "5-knowers", such that they understand the meanings of the numbers 1-5 but not the meaning of any numbers above 5. Nor are there ever any 6 or 7-knowers in children's number learning in industrialized countries. This is a fascinating generalization that has been observed in children learning English, Russian and Japanese. It remains to be known what properties of the child's mind and the input to the child that make this process work the way that it does. The better that we understand this process, the better we can help children acquire their numbers earlier, and be better at arithmetic earlier in life. In order to better understand this process, we studied the progression of number learning in Tsimane' children, a population in a non-industrialized culture, where numeracy information is not particularly valued in the culture. The Tsimane’ live in the tropical rain forest of the Department of Beni, Bolivia. Recent estimates by the Great Tsimane’ Council (the governing body of the Tsimane’) suggest they number approximately 15,000 people living in approximately 120 villages of at least eight households. A typical village has approximately 20 households, with 6 people/household, evenly split between females and males. Subsistence centers on farming and foraging. The Tsimane’ were relatively isolated until the 1970s, when roads brought outsiders into their territory. Contact with the outside is limited to the sale of local goods and work as rural laborers. We recruited 92 children aged 3 to 12 to perform a simple variant of Wynn (1992)'s Give-N task (42 males and 50 females). At the start of testing, a translator explained the task and people should not provide help to the participants since we are interested in what the children know. Each child participant was then asked to move N coins from one half-sheet of white paper to another, with N ranging through a random order of 1, 2, 3, … 8. We defined the following N-knower classification rule for the data from each child (Wynn, 1992): 1. A child is a CP-knower if they make at most one mistake on the 8 trials. 2. A child is an N-knower if they are correct on the first N responses, they never give N for any number words higher than the N'th, and they get at most one other answer correct. 3. A child is a 0-knower if they get at most one answer correct. 4. Otherwise a child cannot be classified into a knower-level. We then tested whether this rule out-performs statistically-matched null data. Note that there are many possible behavioral responses that fail to be classified as a knower level, such as children who know more than one non-sequential number word meaning. We found that 76% of children are assigned to a standard subset-knower level (1-knower through 4-knower or CP-knower). This can be interpreted as roughly the percentage of children who can be "fit" into a knower-level classification. Of the remaining children, 2% are classified as 5-knowers, 0:0% as 6-knowers, and 22% are unclassifiable under our rules. However, examination of the data reveals that many of the children who are counted as "unclassifiable" may actually have a knower-level, and fail to be classified only due to the strictness of our rules. Furthermore, our finding of 76% classification is significantly and substantially above a control, in which the children's responses are shuffled in conservative ways (p < 0:0001). Thus in real data, we are better able to group kids into knower levels than in statistically matched null data with no inherent knower-levels. These results suggest that the trajectory in number word learning reflects a fundamental developmental process. Tsimane' children, like their counterparts in modern, industrialized cultures, successively learn the first three or four words before a conceptual insight that provides all number word meanings. This progression of stages of numerical knowledge is therefore not due to artifacts of education, culture, or age. In fact, our results suggest that this pattern in number learning is a likely developmental universal, to be expected in any place where children must discover how language expresses natural number concepts.
