A dissociation between small and large numbers in young children's ability to "solve for x" in non-symbolic math problems

To add or to remove? The role of working memory updating in preschool children's non-symbolic arithmetic abilities between addition and subtraction

Early computational capacity sets the foundation for mathematical learning. Preschool children have been shown to perform both non-symbolic addition and subtraction problems. However, it is still unknown how different operations affect the representational precision of the non-symbolic arithmetic solutions. The current study compared 83 4- and 5-year-olds' ability to solve non-symbolic addition and subtraction problems and examined the role of working memory underlying the two arithmetic processes. In the task, children were shown two sets of arrays that were sequentially occluded and were asked to either sum the arrays up (addition) or remove one array from the other (subtraction). The solution was then compared with a visible array. Children also completed two working memory tasks to measure their working memory storage and updating abilities. Results showed that children's representational precision in addition was higher than that in subtraction. Although children's performance in both arithmetic operations were associated with working memory updating, solving subtractive problems imposed additional cognitive resources in working memory updating. These findings reveal early developmental differences between addition and subtraction. Children's computational capacity in both addition and subtraction develops early in childhood, and the operation in subtraction demands more mental manipulation in working memory.

What Kinds of Computations Can Young Children Perform Over Non-Symbolic Representations of Small Quantities?

Children can manipulate non-symbolic representations of both small quantities of objects (about four or fewer, represented by the parallel individuation system) and large quantities of objects (represented by the analog magnitude system, or AMS). Previous work has shown that children can perform a variety of non-symbolic operations over AMS representations (like summing and solving for an unknown addend), but are not able to perform further operations on the derived solutions of such non-symbolic operations. However, while the computational capacity of AMS has been studied extensively in early childhood, less is known about the computational capacity of the parallel individuation system. In two experiments, we examined children's ability to perform two types of arithmetic-like operations over representations of small, exact quantities, and whether they could subsequently perform novel operations on derived quantity representations. Four-6-year-old US children (n = 99) solved two types of non-symbolic arithmetic-like problems with small quantities: summation and unknown addend problems. We then tested whether children could use the solutions to these problems as inputs to new operations. Results showed that children more readily solved non-symbolic small, exact addition problems compared to unknown addend problems. However, when children did successfully solve either kind of problem, they were able to use those derived solutions to solve a novel non-symbolic small, exact problem. These results suggest that the parallel individuation system is computationally flexible, contrasting with previous work showing that the AMS is more computationally limited, and shed light on the computational capacities and limitations of representing and operating over representations of small quantities of individual objects.

Development of precision of non-symbolic arithmetic operations in 4-6-year-old children

Children can represent the approximate quantity of sets of items using the Approximate Number System (ANS), and can perform arithmetic-like operations over ANS representations. Previous work has shown that the representational precision of the ANS develops substantially during childhood. However, less is known about the development of the operational precision of the ANS. We examined developmental change in the precision of the solutions to two non-symbolic arithmetic operations in 4-6-year-old U.S. children. We asked children to represent the quantity of an occluded set (Baseline condition), to compute the sum of two sequentially occluded arrays (Addition condition), or to infer the quantity of an addend after observing an initial array and then the array incremented by the unknown addend (Unknown-addend condition). We measured the precision of the solutions of these operations by asking children to compare their solutions to visible arrays, manipulating the ratio between the true quantity of the solution and the comparison array. We found that the precision of ANS representations that were not the result of operations (in the Baseline condition) was higher than the precision of solutions to ANS operations (in the Addition and Unknown-addend conditions). Further, we found that precision in the Baseline and Addition conditions improved significantly between 4 and 6 years, while precision in the Unknown-Addend condition did not. Our results suggest that ANS operations may inject "noise" into the representations they operate over, and that the development of the precision of different operations may follow different trajectories in childhood.

Is Nonsymbolic Arithmetic Truly "Arithmetic"? Examining the Computational Capacity of the Approximate Number System in Young Children

Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic-like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function-like structure, like symbolic arithmetic. Children (n = 74 4- to -8-year-olds in Experiment 1; n = 52 7- to 8-year-olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them "about the same." We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge.

Young Children Intuitively Divide Before They Recognize the Division Symbol

Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.

Children's understanding of economic demand: A dissociation between inference and choice

Adults infer that resources that become scarce over time are in higher demand, and use this "demand inference" to guide their own economic decisions. However, it is unclear when children begin to understand and use economic demand. In six experiments, we investigated the development of demand inference and demand-based economic decisions in 4- to 10-year-old children and adults in the United States. In Experiments 1-5, we showed children two boxes with the same number of compartments but containing different numbers of face-down stickers and varied the information provided about how those differences arose (e.g. that other children had taken the stickers). In separate experiments, we asked children to buy or trade to get a sticker for themselves or to predict what other children would do. We also asked them which set of stickers they thought the other children had preferred to assess their ability to make a demand inference separately from their own choice. Across experiments, children were able to make a demand inference about children's past preferences by 6 years of age. However, children did not use this demand information when making choices for themselves or when predicting what another child would select in the future. In Experiment 6, we adapted the task for adults and found that adult participants inferred that the set containing fewer resources was in higher demand, and selected the higher demand resource for themselves at rates significantly above chance. The overall pattern of results suggests a dissociation between economic inference and economic decisions during early-to-middle childhood. We discuss implications for our understanding of the development of economic reasoning.

First and Second Graders Successfully Reason About Ratios With Both Dot Arrays and Arabic Numerals

Children struggle with exact, symbolic ratio reasoning, but prior research demonstrates children show surprising intuition when making approximate, nonsymbolic ratio judgments. In the current experiment, eighty-five 6- to 8-year-old children made approximate ratio judgments with dot arrays and numerals. Children were adept at approximate ratio reasoning in both formats and improved with age. Children who engaged in the nonsymbolic task first performed better on the symbolic task compared to children tested in the reverse order, suggesting that nonsymbolic ratio reasoning may function as a scaffold for symbolic ratio reasoning. Nonsymbolic ratio reasoning mediated the relation between children's numerosity comparison performance and symbolic mathematics performance in the domain of probabilities, but numerosity comparison performance explained significant unique variance in general numeration skills.